Common Core: High School - Algebra : High School: Algebra

Study concepts, example questions & explanations for Common Core: High School - Algebra

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All Common Core: High School - Algebra Resources

8 Diagnostic Tests 97 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #61 : Arithmetic With Polynomials & Rational Expressions

What is the remainder when \(\displaystyle - 14 x^{2} - 19 x - 1\) is divided by \(\displaystyle x + 15\)

Possible Answers:

\(\displaystyle 192\)

\(\displaystyle -2865\)

\(\displaystyle -33\)

\(\displaystyle 225\)

\(\displaystyle -34\)

Correct answer:

\(\displaystyle -2865\)

Explanation:

In order to solve this problem, we need to perform synthetic division. We set up synthetic division by writing down the zero of the expression we are dividing by, and the coefficients of the polynomial on a line.

\(\displaystyle \begin{tabular}{ c| ccc}-15&-14&-19&-1\\ ~ & ~ & ~ & ~ \\ \hline ~ & ~ & ~ & ~ \end{tabular}\)

The first step is to bring the coefficient of the \(\displaystyle \uptext{x}^2\) term down.

\(\displaystyle \begin{tabular}{ c| ccc}-15&-14&-19&-1\\ ~ & ~ & ~ & ~ \\ \hline ~ &-14& ~ & ~ \end{tabular}\)

Now we multiply the zero by the term we just put down, and place it under the \(\displaystyle \uptext{x}\) term coefficient.

\(\displaystyle \begin{tabular}{ c| ccc}-15&-14&-19&-1\\ ~ & ~ &210& ~ \\ \hline ~ &-14& ~ & ~ \end{tabular}\)

Now we add the column up to get.

\(\displaystyle \begin{tabular}{ c| ccc}-15&-14&-19&-1\\~ & ~ &210& ~ \\ \hline ~ &-14&191& ~ \end{tabular}\)

Now we multiply the number we got by the zero, and place it under the constant term.

\(\displaystyle \begin{tabular}{ c| ccc}-15&-14&-19&-1\\ ~ & ~ &210&-2865\\ \hline ~ &-14&191& ~ \end{tabular}\)

Now we add the column together to get.

\(\displaystyle \begin{tabular}{ c| ccc}-15&-14&-19&-1\\ ~ & ~ &210&-2865\\ \hline ~ &-14&191&-2866\end{tabular}\)

The last number is the remainder, so our final answer is \(\displaystyle -2866\).

Example Question #142 : High School: Algebra

What is the remainder when \(\displaystyle 9 x^{2} + 3 x - 4\) is divided by \(\displaystyle x + 15\)

Possible Answers:

\(\displaystyle 225\)

\(\displaystyle 1980\)

\(\displaystyle 237\)

\(\displaystyle 12\)

\(\displaystyle 8\)

Correct answer:

\(\displaystyle 1980\)

Explanation:

In order to solve this problem, we need to perform synthetic division. We set up synthetic division by writing down the zero of the expression we are dividing by, and the coefficients of the polynomial on a line.

\(\displaystyle \begin{tabular}{ c| ccc}-15&9&3&-4\\ ~ & ~ & ~ & ~ \\ \hline ~ & ~ & ~ & ~ \end{tabular}\)

The first step is to bring the coefficient of the \(\displaystyle \uptext{x}^2\) term down.

\(\displaystyle \begin{tabular}{ c| ccc}-15&9&3&-4\\ ~ & ~ & ~ & ~ \\ \hline ~ &9& ~ & ~ \end{tabular}\)

Now we multiply the zero by the term we just put down, and place it under the \(\displaystyle \uptext{x}\) term coefficient.

\(\displaystyle \begin{tabular}{ c| ccc}-15&9&3&-4\\ ~ & ~ &-135& ~ \\ \hline ~ &9& ~ & ~ \end{tabular}\)

Now we add the column up to get.

\(\displaystyle \begin{tabular}{ c| ccc}-15&9&3&-4\\~ & ~ &-135& ~ \\ \hline ~ &9&-132& ~ \end{tabular}\)

Now we multiply the number we got by the zero, and place it under the constant term.

\(\displaystyle \begin{tabular}{ c| ccc}-15&9&3&-4\\ ~ & ~ &-135&1980\\ \hline ~ &9&-132& ~ \end{tabular}\)

Now we add the column together to get.

\(\displaystyle \begin{tabular}{ c| ccc}-15&9&3&-4\\ ~ & ~ &-135&1980\\ \hline ~ &9&-132&1976\end{tabular}\)

The last number is the remainder, so our final answer is \(\displaystyle 1976\).

Example Question #71 : Arithmetic With Polynomials & Rational Expressions

What is the remainder when \(\displaystyle - 11 x^{2} - 10 x + 9\) is divided by \(\displaystyle x - 18\)

Possible Answers:

\(\displaystyle 303\)

\(\displaystyle -3744\)

\(\displaystyle 324\)

\(\displaystyle -21\)

\(\displaystyle -12\)

Correct answer:

\(\displaystyle -3744\)

Explanation:

In order to solve this problem, we need to perform synthetic division. We set up synthetic division by writing down the zero of the expression we are dividing by, and the coefficients of the polynomial on a line.

\(\displaystyle \begin{tabular}{ c| ccc}18&-11&-10&9\\ ~ & ~ & ~ & ~ \\ \hline ~ & ~ & ~ & ~ \end{tabular}\)

The first step is to bring the coefficient of the \(\displaystyle \uptext{x}^2\) term down.

\(\displaystyle \begin{tabular}{ c| ccc}18&-11&-10&9\\ ~ & ~ & ~ & ~ \\ \hline ~ &-11& ~ & ~ \end{tabular}\)

Now we multiply the zero by the term we just put down, and place it under the \(\displaystyle \uptext{x}\) term coefficient.

\(\displaystyle \begin{tabular}{ c| ccc}18&-11&-10&9\\ ~ & ~ &-198& ~ \\ \hline ~ &-11& ~ & ~ \end{tabular}\)

Now we add the column up to get.

\(\displaystyle \begin{tabular}{ c| ccc}18&-11&-10&9\\~ & ~ &-198& ~ \\ \hline ~ &-11&-208& ~ \end{tabular}\)

Now we multiply the number we got by the zero, and place it under the constant term.

\(\displaystyle \begin{tabular}{ c| ccc}18&-11&-10&9\\ ~ & ~ &-198&-3744\\ \hline ~ &-11&-208& ~ \end{tabular}\)

Now we add the column together to get.

\(\displaystyle \begin{tabular}{ c| ccc}18&-11&-10&9\\ ~ & ~ &-198&-3744\\ \hline ~ &-11&-208&-3735\end{tabular}\)

The last number is the remainder, so our final answer is \(\displaystyle -3735\).

Example Question #72 : Arithmetic With Polynomials & Rational Expressions

What is the remainder when \(\displaystyle - 5 x^{2} + 10 x - 7\) is divided by \(\displaystyle x + 17\)

Possible Answers:

\(\displaystyle 5\)

\(\displaystyle 289\)

\(\displaystyle 294\)

\(\displaystyle -1615\)

\(\displaystyle -2\)

Correct answer:

\(\displaystyle -1615\)

Explanation:

In order to solve this problem, we need to perform synthetic division. We set up synthetic division by writing down the zero of the expression we are dividing by, and the coefficients of the polynomial on a line.

\(\displaystyle \begin{tabular}{ c| ccc}-17&-5&10&-7\\ ~ & ~ & ~ & ~ \\ \hline ~ & ~ & ~ & ~ \end{tabular}\)

The first step is to bring the coefficient of the \(\displaystyle \uptext{x}^2\) term down.

\(\displaystyle \begin{tabular}{ c| ccc}-17&-5&10&-7\\ ~ & ~ & ~ & ~ \\ \hline ~ &-5& ~ & ~ \end{tabular}\)

Now we multiply the zero by the term we just put down, and place it under the \(\displaystyle \uptext{x}\) term coefficient.

\(\displaystyle \begin{tabular}{ c| ccc}-17&-5&10&-7\\ ~ & ~ &85& ~ \\ \hline ~ &-5& ~ & ~ \end{tabular}\)

Now we add the column up to get.

\(\displaystyle \begin{tabular}{ c| ccc}-17&-5&10&-7\\~ & ~ &85& ~ \\ \hline ~ &-5&95& ~ \end{tabular}\)

Now we multiply the number we got by the zero, and place it under the constant term.

\(\displaystyle \begin{tabular}{ c| ccc}-17&-5&10&-7\\ ~ & ~ &85&-1615\\ \hline ~ &-5&95& ~ \end{tabular}\)

Now we add the column together to get.

\(\displaystyle \begin{tabular}{ c| ccc}-17&-5&10&-7\\ ~ & ~ &85&-1615\\ \hline ~ &-5&95&-1622\end{tabular}\)

The last number is the remainder, so our final answer is \(\displaystyle -1622\).

Example Question #61 : Remainder Theorem: Ccss.Math.Content.Hsa Apr.B.2

What is the remainder when \(\displaystyle 9 x^{2} - 7 x + 7\) is divided by \(\displaystyle x + 12\)

Possible Answers:

\(\displaystyle 9\)

\(\displaystyle 146\)

\(\displaystyle 144\)

\(\displaystyle 2\)

\(\displaystyle 1380\)

Correct answer:

\(\displaystyle 1380\)

Explanation:

In order to solve this problem, we need to perform synthetic division. We set up synthetic division by writing down the zero of the expression we are dividing by, and the coefficients of the polynomial on a line.

\(\displaystyle \begin{tabular}{ c| ccc}-12&9&-7&7\\ ~ & ~ & ~ & ~ \\ \hline ~ & ~ & ~ & ~ \end{tabular}\)

The first step is to bring the coefficient of the \(\displaystyle \uptext{x}^2\) term down.

\(\displaystyle \begin{tabular}{ c| ccc}-12&9&-7&7\\ ~ & ~ & ~ & ~ \\ \hline ~ &9& ~ & ~ \end{tabular}\)

Now we multiply the zero by the term we just put down, and place it under the \(\displaystyle \uptext{x}\) term coefficient.

\(\displaystyle \begin{tabular}{ c| ccc}-12&9&-7&7\\ ~ & ~ &-108& ~ \\ \hline ~ &9& ~ & ~ \end{tabular}\)

Now we add the column up to get.

\(\displaystyle \begin{tabular}{ c| ccc}-12&9&-7&7\\~ & ~ &-108& ~ \\ \hline ~ &9&-115& ~ \end{tabular}\)

Now we multiply the number we got by the zero, and place it under the constant term.

\(\displaystyle \begin{tabular}{ c| ccc}-12&9&-7&7\\ ~ & ~ &-108&1380\\ \hline ~ &9&-115& ~ \end{tabular}\)

Now we add the column together to get.

\(\displaystyle \begin{tabular}{ c| ccc}-12&9&-7&7\\ ~ & ~ &-108&1380\\ \hline ~ &9&-115&1387\end{tabular}\)

The last number is the remainder, so our final answer is \(\displaystyle 1387\).

Example Question #62 : Remainder Theorem: Ccss.Math.Content.Hsa Apr.B.2

What is the remainder when \(\displaystyle - 9 x^{2} - 9 x - 4\) is divided by \(\displaystyle x + 11\)

Possible Answers:

\(\displaystyle -990\)

\(\displaystyle 103\)

\(\displaystyle -18\)

\(\displaystyle -22\)

\(\displaystyle 121\)

Correct answer:

\(\displaystyle -990\)

Explanation:

In order to solve this problem, we need to perform synthetic division. We set up synthetic division by writing down the zero of the expression we are dividing by, and the coefficients of the polynomial on a line.

\(\displaystyle \begin{tabular}{ c| ccc}-11&-9&-9&-4\\ ~ & ~ & ~ & ~ \\ \hline ~ & ~ & ~ & ~ \end{tabular}\)

The first step is to bring the coefficient of the \(\displaystyle \uptext{x}^2\) term down.

\(\displaystyle \begin{tabular}{ c| ccc}-11&-9&-9&-4\\ ~ & ~ & ~ & ~ \\ \hline ~ &-9& ~ & ~ \end{tabular}\)

Now we multiply the zero by the term we just put down, and place it under the \(\displaystyle \uptext{x}\) term coefficient.

\(\displaystyle \begin{tabular}{ c| ccc}-11&-9&-9&-4\\ ~ & ~ &99& ~ \\ \hline ~ &-9& ~ & ~ \end{tabular}\)

Now we add the column up to get.

\(\displaystyle \begin{tabular}{ c| ccc}-11&-9&-9&-4\\~ & ~ &99& ~ \\ \hline ~ &-9&90& ~ \end{tabular}\)

Now we multiply the number we got by the zero, and place it under the constant term.

\(\displaystyle \begin{tabular}{ c| ccc}-11&-9&-9&-4\\ ~ & ~ &99&-990\\ \hline ~ &-9&90& ~ \end{tabular}\)

Now we add the column together to get.

\(\displaystyle \begin{tabular}{ c| ccc}-11&-9&-9&-4\\ ~ & ~ &99&-990\\ \hline ~ &-9&90&-994\end{tabular}\)

The last number is the remainder, so our final answer is \(\displaystyle -994\).

Example Question #63 : Remainder Theorem: Ccss.Math.Content.Hsa Apr.B.2

What is the remainder when \(\displaystyle - 13 x^{2} - 8 x + 7\) is divided by \(\displaystyle x + 14\)

Possible Answers:

\(\displaystyle -14\)

\(\displaystyle 196\)

\(\displaystyle 175\)

\(\displaystyle -21\)

\(\displaystyle -2436\)

Correct answer:

\(\displaystyle -2436\)

Explanation:

In order to solve this problem, we need to perform synthetic division. We set up synthetic division by writing down the zero of the expression we are dividing by, and the coefficients of the polynomial on a line.

\(\displaystyle \begin{tabular}{ c| ccc}-14&-13&-8&7\\ ~ & ~ & ~ & ~ \\ \hline ~ & ~ & ~ & ~ \end{tabular}\)

The first step is to bring the coefficient of the \(\displaystyle \uptext{x}^2\) term down.

\(\displaystyle \begin{tabular}{ c| ccc}-14&-13&-8&7\\ ~ & ~ & ~ & ~ \\ \hline ~ &-13& ~ & ~ \end{tabular}\)

Now we multiply the zero by the term we just put down, and place it under the \(\displaystyle \uptext{x}\) term coefficient.

\(\displaystyle \begin{tabular}{ c| ccc}-14&-13&-8&7\\ ~ & ~ &182& ~ \\ \hline ~ &-13& ~ & ~ \end{tabular}\)

Now we add the column up to get.

\(\displaystyle \begin{tabular}{ c| ccc}-14&-13&-8&7\\~ & ~ &182& ~ \\ \hline ~ &-13&174& ~ \end{tabular}\)

Now we multiply the number we got by the zero, and place it under the constant term.

\(\displaystyle \begin{tabular}{ c| ccc}-14&-13&-8&7\\ ~ & ~ &182&-2436\\ \hline ~ &-13&174& ~ \end{tabular}\)

Now we add the column together to get.

\(\displaystyle \begin{tabular}{ c| ccc}-14&-13&-8&7\\ ~ & ~ &182&-2436\\ \hline ~ &-13&174&-2429\end{tabular}\)

The last number is the remainder, so our final answer is \(\displaystyle -2429\).

Example Question #64 : Remainder Theorem: Ccss.Math.Content.Hsa Apr.B.2

What is the remainder when \(\displaystyle 19 x^{2} - 11 x - 2\) is divided by \(\displaystyle x + 14\)

Possible Answers:

\(\displaystyle 3878\)

\(\displaystyle 8\)

\(\displaystyle 196\)

\(\displaystyle 204\)

\(\displaystyle 6\)

Correct answer:

\(\displaystyle 3878\)

Explanation:

In order to solve this problem, we need to perform synthetic division. We set up synthetic division by writing down the zero of the expression we are dividing by, and the coefficients of the polynomial on a line.

\(\displaystyle \begin{tabular}{ c| ccc}-14&19&-11&-2\\ ~ & ~ & ~ & ~ \\ \hline ~ & ~ & ~ & ~ \end{tabular}\)

The first step is to bring the coefficient of the \(\displaystyle \uptext{x}^2\) term down.

\(\displaystyle \begin{tabular}{ c| ccc}-14&19&-11&-2\\ ~ & ~ & ~ & ~ \\ \hline ~ &19& ~ & ~ \end{tabular}\)

Now we multiply the zero by the term we just put down, and place it under the \(\displaystyle \uptext{x}\) term coefficient.

\(\displaystyle \begin{tabular}{ c| ccc}-14&19&-11&-2\\ ~ & ~ &-266& ~ \\ \hline ~ &19& ~ & ~ \end{tabular}\)

Now we add the column up to get.

\(\displaystyle \begin{tabular}{ c| ccc}-14&19&-11&-2\\~ & ~ &-266& ~ \\ \hline ~ &19&-277& ~ \end{tabular}\)

Now we multiply the number we got by the zero, and place it under the constant term.

\(\displaystyle \begin{tabular}{ c| ccc}-14&19&-11&-2\\ ~ & ~ &-266&3878\\ \hline ~ &19&-277& ~ \end{tabular}\)

Now we add the column together to get.

\(\displaystyle \begin{tabular}{ c| ccc}-14&19&-11&-2\\ ~ & ~ &-266&3878\\ \hline ~ &19&-277&3876\end{tabular}\)

The last number is the remainder, so our final answer is \(\displaystyle 3876\).

Example Question #65 : Remainder Theorem: Ccss.Math.Content.Hsa Apr.B.2

What is the remainder when \(\displaystyle - 6 x^{2} - 4 x - 3\) is divided by \(\displaystyle x + 8\)

Possible Answers:

\(\displaystyle -352\)

\(\displaystyle -13\)

\(\displaystyle 54\)

\(\displaystyle 64\)

\(\displaystyle -10\)

Correct answer:

\(\displaystyle -352\)

Explanation:

In order to solve this problem, we need to perform synthetic division. We set up synthetic division by writing down the zero of the expression we are dividing by, and the coefficients of the polynomial on a line.

\(\displaystyle \begin{tabular}{ c| ccc}-8&-6&-4&-3\\ ~ & ~ & ~ & ~ \\ \hline ~ & ~ & ~ & ~ \end{tabular}\)

The first step is to bring the coefficient of the \(\displaystyle \uptext{x}^2\) term down.

\(\displaystyle \begin{tabular}{ c| ccc}-8&-6&-4&-3\\ ~ & ~ & ~ & ~ \\ \hline ~ &-6& ~ & ~ \end{tabular}\)

Now we multiply the zero by the term we just put down, and place it under the \(\displaystyle \uptext{x}\) term coefficient.

\(\displaystyle \begin{tabular}{ c| ccc}-8&-6&-4&-3\\ ~ & ~ &48& ~ \\ \hline ~ &-6& ~ & ~ \end{tabular}\)

Now we add the column up to get.

\(\displaystyle \begin{tabular}{ c| ccc}-8&-6&-4&-3\\~ & ~ &48& ~ \\ \hline ~ &-6&44& ~ \end{tabular}\)

Now we multiply the number we got by the zero, and place it under the constant term.

\(\displaystyle \begin{tabular}{ c| ccc}-8&-6&-4&-3\\ ~ & ~ &48&-352\\ \hline ~ &-6&44& ~ \end{tabular}\)

Now we add the column together to get.

\(\displaystyle \begin{tabular}{ c| ccc}-8&-6&-4&-3\\ ~ & ~ &48&-352\\ \hline ~ &-6&44&-355\end{tabular}\)

The last number is the remainder, so our final answer is \(\displaystyle -355\).

Example Question #66 : Remainder Theorem: Ccss.Math.Content.Hsa Apr.B.2

What is the remainder when \(\displaystyle - 3 x^{2} - 8 x - 4\) is divided by \(\displaystyle x + 5\)

Possible Answers:

\(\displaystyle -35\)

\(\displaystyle -15\)

\(\displaystyle 25\)

\(\displaystyle 14\)

\(\displaystyle -11\)

Correct answer:

\(\displaystyle -35\)

Explanation:

In order to solve this problem, we need to perform synthetic division. We set up synthetic division by writing down the zero of the expression we are dividing by, and the coefficients of the polynomial on a line.

\(\displaystyle \begin{tabular}{ c| ccc}-5&-3&-8&-4\\ ~ & ~ & ~ & ~ \\ \hline ~ & ~ & ~ & ~ \end{tabular}\)

The first step is to bring the coefficient of the \(\displaystyle \uptext{x}^2\) term down.

\(\displaystyle \begin{tabular}{ c| ccc}-5&-3&-8&-4\\ ~ & ~ & ~ & ~ \\ \hline ~ &-3& ~ & ~ \end{tabular}\)

Now we multiply the zero by the term we just put down, and place it under the \(\displaystyle \uptext{x}\) term coefficient.

\(\displaystyle \begin{tabular}{ c| ccc}-5&-3&-8&-4\\ ~ & ~ &15& ~ \\ \hline ~ &-3& ~ & ~ \end{tabular}\)

Now we add the column up to get.

\(\displaystyle \begin{tabular}{ c| ccc}-5&-3&-8&-4\\~ & ~ &15& ~ \\ \hline ~ &-3&7& ~ \end{tabular}\)

Now we multiply the number we got by the zero, and place it under the constant term.

\(\displaystyle \begin{tabular}{ c| ccc}-5&-3&-8&-4\\ ~ & ~ &15&-35\\ \hline ~ &-3&7& ~ \end{tabular}\)

Now we add the column together to get.

\(\displaystyle \begin{tabular}{ c| ccc}-5&-3&-8&-4\\ ~ & ~ &15&-35\\ \hline ~ &-3&7&-39\end{tabular}\)

The last number is the remainder, so our final answer is \(\displaystyle -39\).

All Common Core: High School - Algebra Resources

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