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Common Core: High School - Algebra : Arithmetic with Polynomials & Rational Expressions

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

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Example Questions

Example Question #1 : Polynomials And Quadratics

Given $\displaystyle A$ and $\displaystyle B$ find A-B $\displaystyle A-B$ .

$\\A=x^2+2 \\B=4x^2-1$ $\displaystyle \\A=x^2+2 \\B=4x^2-1$

Possible Answers:

-3x^2-3 $\displaystyle -3x^2-3$

-3x^2+3 $\displaystyle -3x^2+3$

3x^2+3 $\displaystyle 3x^2+3$

-5x^2+3 $\displaystyle -5x^2+3$

3x^2-3 $\displaystyle 3x^2-3$

Correct answer:

-3x^2+3 $\displaystyle -3x^2+3$

Explanation:

To find the difference of two polynomials first set up the operation and identify the like terms.

$\\A=x^2+2 \\B=4x^2-1$ $\displaystyle \\A=x^2+2 \\B=4x^2-1$

$\displaystyle A-B= (x^2+2)-(4x^2-1)$

The like terms in these polynomials are the squared variable and the constant terms.

Remember to distribute the negative sign through to all terms in the second polynomial.

$A-B={\color{Red} x^2}{\color{Blue} +2}{\color{Red} -4x^2}{\color{Blue} --1}$ $\displaystyle A-B={\color{Red} x^2}{\color{Blue} +2}{\color{Red} -4x^2}{\color{Blue} --1}$

$\\x^2-4x^2=-3x^2 \\2-(-1)=3$ $\displaystyle \\x^2-4x^2=-3x^2 \\2-(-1)=3$

Therefore, the sum of these polynomials is,

-3x^2+3 $\displaystyle -3x^2+3$

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Example Question #2 : Polynomials And Quadratics

Given $\displaystyle A$ and $\displaystyle B$ find A+B $\displaystyle A+B$ .

$\\A=x^2+2 \\B=4x^2-1$ $\displaystyle \\A=x^2+2 \\B=4x^2-1$

Possible Answers:

5x^2-1 $\displaystyle 5x^2-1$

5x^2+3 $\displaystyle 5x^2+3$

5x^2+1 $\displaystyle 5x^2+1$

3x^2+1 $\displaystyle 3x^2+1$

3x^2-1 $\displaystyle 3x^2-1$

Correct answer:

5x^2+1 $\displaystyle 5x^2+1$

Explanation:

To find the sum of two polynomials first set up the operation and identify the like terms.

$\\A=x^2+2 \\B=4x^2-1$ $\displaystyle \\A=x^2+2 \\B=4x^2-1$

$\displaystyle A+B= (x^2+2)+(4x^2-1)$

The like terms in these polynomials are the squared variable and the constant terms.

$A+B={\color{Red} x^2}{\color{Blue} +2}{\color{Red} +4x^2}{\color{Blue} -1}$ $\displaystyle A+B={\color{Red} x^2}{\color{Blue} +2}{\color{Red} +4x^2}{\color{Blue} -1}$

$\\x^2+4x^2=5x^2 \\2+(-1)=1$ $\displaystyle \\x^2+4x^2=5x^2 \\2+(-1)=1$

Therefore, the sum of these polynomials is,

5x^2+1 $\displaystyle 5x^2+1$

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Example Question #3 : Polynomials And Quadratics

Given $\displaystyle A$ and $\displaystyle B$ find B-A $\displaystyle B-A$ .

$\\A=x^2+2 \\B=4x^2-1$ $\displaystyle \\A=x^2+2 \\B=4x^2-1$

Possible Answers:

-3x^2-3 $\displaystyle -3x^2-3$

3x^2-3 $\displaystyle 3x^2-3$

5x^2-3 $\displaystyle 5x^2-3$

-3x^2+3 $\displaystyle -3x^2+3$

3x^2+3 $\displaystyle 3x^2+3$

Correct answer:

3x^2-3 $\displaystyle 3x^2-3$

Explanation:

To find the difference of two polynomials first set up the operation and identify the like terms.

$\\A=x^2+2 \\B=4x^2-1$ $\displaystyle \\A=x^2+2 \\B=4x^2-1$

$\displaystyle B-A= (4x^2-1)-(x^2+2)$

The like terms in these polynomials are the squared variable and the constant terms.

Remember to distribute the negative sign through to all terms in the second polynomial.

$B-A={\color{Red} 4x^2}{\color{Blue} -1}{\color{Red} -x^2}{\color{Blue} -2}$ $\displaystyle B-A={\color{Red} 4x^2}{\color{Blue} -1}{\color{Red} -x^2}{\color{Blue} -2}$

$\\4x^2-x^2=3x^2 \\-1-2=-3$ $\displaystyle \\4x^2-x^2=3x^2 \\-1-2=-3$

Therefore, the sum of these polynomials is,

3x^2-3 $\displaystyle 3x^2-3$

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Example Question #4 : Polynomials And Quadratics

Given $\displaystyle A$ and $\displaystyle B$ find A-B $\displaystyle A-B$ .

$\\A=3x^3+x+2 \\B=4x^3-2x-1$ $\displaystyle \\A=3x^3+x+2 \\B=4x^3-2x-1$

Possible Answers:

$\displaystyle x^3+3x+3$

$\displaystyle -x^3-3x-3$

$\displaystyle x^3+3x-3$

$\displaystyle -x^3+3x+3$

$\displaystyle -x^3-3x+3$

Correct answer:

$\displaystyle -x^3+3x+3$

Explanation:

To find the difference of two polynomials first set up the operation and identify the like terms.

$\\A=3x^3+x+2 \\B=4x^3-2x-1$ $\displaystyle \\A=3x^3+x+2 \\B=4x^3-2x-1$

$\displaystyle A-B= (3x^3+x+2)-(4x^3-2x-1)$

The like terms in these polynomials are the squared variable, the single variable, and the constant terms.

Remember to distribute the negative sign to all terms within the parentheses.

$A-B={\color{Red} 3x^3}{\color{Blue} +x}+2{\color{Red} -4x^3}{\color{Blue} +2x}+1$ $\displaystyle A-B={\color{Red} 3x^3}{\color{Blue} +x}+2{\color{Red} -4x^3}{\color{Blue} +2x}+1$

$\\3x^3-4x^3=-x^3 \\x+2x=3x \\2+1=3$ $\displaystyle \\3x^3-4x^3=-x^3 \\x+2x=3x \\2+1=3$

Therefore, the sum of these polynomials is,

$\displaystyle -x^3+3x+3$

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Example Question #5 : Polynomials And Quadratics

Given $\displaystyle A$ and $\displaystyle B$ find A+B $\displaystyle A+B$ .

$\\A=2x^2+2 \\B=4x^2-2$ $\displaystyle \\A=2x^2+2 \\B=4x^2-2$

Possible Answers:

6x^2+2 $\displaystyle 6x^2+2$

6x^2+4 $\displaystyle 6x^2+4$

6x^2 $\displaystyle 6x^2$

-6x^2 $\displaystyle -6x^2$

6x^2-4 $\displaystyle 6x^2-4$

Correct answer:

6x^2 $\displaystyle 6x^2$

Explanation:

To find the sum of two polynomials first set up the operation and identify the like terms.

$\\A=2x^2+2 \\B=4x^2-2$ $\displaystyle \\A=2x^2+2 \\B=4x^2-2$

$\displaystyle A+B= (2x^2+2)+(4x^2-2)$

The like terms in these polynomials are the squared variable and the constant terms.

$A+B={\color{Red} 2x^2}{\color{Blue} +2}{\color{Red} +4x^2}{\color{Blue} -2}$ $\displaystyle A+B={\color{Red} 2x^2}{\color{Blue} +2}{\color{Red} +4x^2}{\color{Blue} -2}$

$\\2x^2+4x^2=6x^2 \\2+(-2)=0$ $\displaystyle \\2x^2+4x^2=6x^2 \\2+(-2)=0$

Therefore, the sum of these polynomials is,

6x^2 $\displaystyle 6x^2$

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Example Question #6 : Polynomials And Quadratics

Given $\displaystyle A$ and $\displaystyle B$ find A-B $\displaystyle A-B$ .

$\\A=2x^2+2 \\B=4x^2-2$ $\displaystyle \\A=2x^2+2 \\B=4x^2-2$

Possible Answers:

2x^2+4 $\displaystyle 2x^2+4$

-2x^2-4 $\displaystyle -2x^2-4$

-2x^2+4 $\displaystyle -2x^2+4$

2x^2-4 $\displaystyle 2x^2-4$

-2x^2+2 $\displaystyle -2x^2+2$

Correct answer:

-2x^2+4 $\displaystyle -2x^2+4$

Explanation:

To find the difference of two polynomials first set up the operation and identify the like terms.

$\\A=2x^2+2 \\B=4x^2-2$ $\displaystyle \\A=2x^2+2 \\B=4x^2-2$

$\displaystyle A-B= (2x^2+2)-(4x^2-2)$

The like terms in these polynomials are the squared variable and the constant terms.

Remember to distribute the negative sign to all terms in the second polynomial.

$A-B={\color{Red} 2x^2}{\color{Blue} +2}{\color{Red} -4x^2}{\color{Blue} +2}$ $\displaystyle A-B={\color{Red} 2x^2}{\color{Blue} +2}{\color{Red} -4x^2}{\color{Blue} +2}$

$\\2x^2-4x^2=-2x^2 \\2+2=4$ $\displaystyle \\2x^2-4x^2=-2x^2 \\2+2=4$

Therefore, the sum of these polynomials is,

-2x^2+4 $\displaystyle -2x^2+4$

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Example Question #7 : Polynomials And Quadratics

Given $\displaystyle A$ and $\displaystyle B$ find B-A $\displaystyle B-A$ .

$\\A=2x^2+2 \\B=4x^2-2$ $\displaystyle \\A=2x^2+2 \\B=4x^2-2$

Possible Answers:

2x^2 $\displaystyle 2x^2$

2x^2-4 $\displaystyle 2x^2-4$

-2x^2-4 $\displaystyle -2x^2-4$

2x^2+4 $\displaystyle 2x^2+4$

-2x^2+4 $\displaystyle -2x^2+4$

Correct answer:

2x^2-4 $\displaystyle 2x^2-4$

Explanation:

To find the difference of two polynomials first set up the operation and identify the like terms.

$\\A=2x^2+2 \\B=4x^2-2$ $\displaystyle \\A=2x^2+2 \\B=4x^2-2$

$\displaystyle B-A= (4x^2-2)-(2x^2+2)$

The like terms in these polynomials are the squared variable and the constant terms.

Remember to distribute the negative sign to all terms in the second polynomial.

$B-A= {\color{Red} 4x^2}-2{\color{Red} -2x^2}-2$ $\displaystyle B-A= {\color{Red} 4x^2}-2{\color{Red} -2x^2}-2$

$\\4x^2-2x^2=2x^2 \\-2-2=-4$ $\displaystyle \\4x^2-2x^2=2x^2 \\-2-2=-4$

Therefore, the sum of these polynomials is,

2x^2-4 $\displaystyle 2x^2-4$

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Example Question #8 : Polynomials And Quadratics

Given $\displaystyle A$ and $\displaystyle B$ find A+B $\displaystyle A+B$ .

$\\A=x^2 \\B=4x^2-1$ $\displaystyle \\A=x^2 \\B=4x^2-1$

Possible Answers:

5x^2-1 $\displaystyle 5x^2-1$

-5x^2+1 $\displaystyle -5x^2+1$

5x^2+1 $\displaystyle 5x^2+1$

5x^2 $\displaystyle 5x^2$

-5x^2-1 $\displaystyle -5x^2-1$

Correct answer:

5x^2-1 $\displaystyle 5x^2-1$

Explanation:

To find the sum of two polynomials first set up the operation and identify the like terms.

$\\A=x^2 \\B=4x^2-1$ $\displaystyle \\A=x^2 \\B=4x^2-1$

$\displaystyle A+B= (x^2)+(4x^2-1)$

The like terms in these polynomials are the squared variable.

$A+B={\color{Red} x^2}{\color{Red} +4x^2}-1$ $\displaystyle A+B={\color{Red} x^2}{\color{Red} +4x^2}-1$

$\\x^2+4x^2=5x^2 \\0+(-1)=-1$ $\displaystyle \\x^2+4x^2=5x^2 \\0+(-1)=-1$

Therefore, the sum of these polynomials is,

5x^2-1 $\displaystyle 5x^2-1$

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Example Question #9 : Polynomials And Quadratics

Given $\displaystyle A$ and $\displaystyle B$ find $\displaystyle AB$ .

$\\A=x^2+1 \\B=4x^2-1$ $\displaystyle \\A=x^2+1 \\B=4x^2-1$

Possible Answers:

$\displaystyle -4x^4-3x^2-1$

$\displaystyle 4x^4-3x^2-1$

$\displaystyle 4x^4+3x^2-1$

$\displaystyle 4x^4+3x^2+1$

$\displaystyle -4x^4+3x^2-1$

Correct answer:

$\displaystyle 4x^4+3x^2-1$

Explanation:

To find the product of two polynomials first set up the operation.

$\\A=x^2+1 \\B=4x^2-1$ $\displaystyle \\A=x^2+1 \\B=4x^2-1$

$\displaystyle AB=(x^2+1)(4x^2-1)$

Now, multiply each term from the first polynomial with each term in the second polynomial.

Remember the rules of exponents. When like base variables are multiplied together their exponents are added together.

$\\x^2\cdot 4x^2=4x^{2+2}=4x^4 \\x^2\cdot -1=-x^2 \\1\cdot 4x^2=4x^2 \\1\cdot -1=-1$ $\displaystyle \\x^2\cdot 4x^2=4x^{2+2}=4x^4 \\x^2\cdot -1=-x^2 \\1\cdot 4x^2=4x^2 \\1\cdot -1=-1$

Therefore, the product of these polynomials is,

$\displaystyle 4x^4-x^2+4x^2-1$

Combine like terms to arrive at the final answer.

$\displaystyle 4x^4+3x^2-1$

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Example Question #10 : Polynomials And Quadratics

If $A=3x^{3}+x+2$ $\displaystyle A=3x^{3}+x+2$ , and $B=4x^{3}-2x-1$ $\displaystyle B=4x^{3}-2x-1$ , what is the value of A+B? $\displaystyle A+B?$

Possible Answers:

$\displaystyle 7x^3+x-1$

$\displaystyle 7x^3-x-1$

$\displaystyle -7x^3-x+1$

$\displaystyle 7x^3+x+1$

$\displaystyle 7x^3-x+1$

Correct answer:

$\displaystyle 7x^3-x+1$

Explanation:

In order to find the sum of two polynomials, we must first set up the operation and identify the like terms.

$\\A=3x^3+x+2 \\B=4x^3-2x-1$ $\displaystyle \\A=3x^3+x+2 \\B=4x^3-2x-1$

$\displaystyle A+B= (3x^3+x+2)+(4x^3-2x-1)$

The like terms in these polynomials are the squared variable, the single variable, and the constant terms.

$\displaystyle A+B= 3x^3+x+2+4x^3-2x-1$

$\\3x^3+4x^3=7x^3 \\x-2x=-x \\2-1=1$ $\displaystyle \\3x^3+4x^3=7x^3 \\x-2x=-x \\2-1=1$

The sum of these polynomials is equal to the following expression:

$\displaystyle 7x^3-x+1$

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Common Core: High School - Algebra : Arithmetic with Polynomials & Rational Expressions

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

All Common Core: High School - Algebra Resources

Example Questions

Example Question #1 : Polynomials And Quadratics

Example Question #2 : Polynomials And Quadratics

Example Question #3 : Polynomials And Quadratics

Example Question #4 : Polynomials And Quadratics

Example Question #5 : Polynomials And Quadratics

Example Question #6 : Polynomials And Quadratics

Example Question #7 : Polynomials And Quadratics

Example Question #8 : Polynomials And Quadratics

Example Question #9 : Polynomials And Quadratics

Example Question #10 : Polynomials And Quadratics

All Common Core: High School - Algebra Resources

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