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Algebra 1 : Polynomial Operations

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Example Question #51 : Polynomial Operations

Solve:

$\frac{5}{x+1}-\frac{3x}{\left ( x+1 \right )\left ( x-2 \right )}= \frac{3}{x-2}$

Possible Answers:

$\frac{13}{2}$

Correct answer:

Explanation:

First we convert each of the denominators into an LCD which gives us the following:

$\frac{5\left ( x-2 \right )}{\left ( x+1 \right )\left ( x-2 \right )}- \frac{3x}{\left ( x+1 \right )\left ( x-2 \right )}= \frac{3\left ( x+1 \right )}{\left ( x+1 \right )\left ( x-2 \right )}$

Now we add or subtract the numerators which gives us:

$5\left ( x-2 \right )-3x= 3\left ( x+1 \right )$

Simplifying the above equation gives us the answer which is:

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Example Question #1 : How To Subtract Polynomials

Simplify the following:

4x^3-2x(x-2)(x+3)

Possible Answers:

4x^3-2x(x^2+x-6)

4x^3-2x^3-2x^2+12x

2x^3-2x^2+12x

8x^3-2x+16

None of the other answers are correct.

Correct answer:

2x^3-2x^2+12x

Explanation:

4x^3-2x(x-2)(x+3)

First, FOIL the two binomials:

4x^3-2x(x^2+x-6)

Then distribute the through the terms in parentheses:

4x^3-2x^3-2x^2+12x

Combine like terms:

2x^3-2x^2+12x

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Example Question #3 : How To Subtract Polynomials

Simplify.

$(4a^{2}b+2ab) - (3a^{2}b+4ab+2)$

Possible Answers:

$a^{2}b-6ab-2$

$a^{2}b-2ab-2$

$a^{2}b+6ab+2$

$a^{2}b+2ab-2$

Correct answer:

$a^{2}b-2ab-2$

Explanation:

Simplify

$(4a^{2}b+2ab) - (3a^{2}b+4ab+2)$

Distribute the negative: $\boldsymbol{-}3a^{2}b\boldsymbol{-}4ab\boldsymbol{-}2$

Then combinde like terms

$\mathbf{\boldsymbol{-}3a^{2}b+4a^{2}b}(\mathbf{\boldsymbol{-}4ab+2ab})\boldsymbol{-}2$

Answer:

$a^{2}b-2ab-2$

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Example Question #4 : How To Subtract Polynomials

Subtract:

$(7.5x^{2} + 1.4x - 4.6) - (x^{2} + 6.2x - 3.5)$

Possible Answers:

$6.5x^{2} +7.2x - 1.1$

$6.5x^{2} - 4.8x - 8.1$

$6.5x^{2} - 4.8x - 1.1$

$7.5x^{2} +7.2x - 1.1$

$7.5x^{2} - 4.8x - 1.1$

Correct answer:

$6.5x^{2} - 4.8x - 1.1$

Explanation:

$(7.5x^{2} + 1.4x - 4.6) - (x^{2} + 6.2x - 3.5)$

$= 7.5x^{2} + 1.4x - 4.6 - x^{2} - 6.2x + 3.5$

$= 7.5x^{2} - 1x^{2} + 1.4x- 6.2x - 4.6 + 3.5$

$= 6.5x^{2} - 4.8x - 1.1$

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Example Question #5 : How To Subtract Polynomials

Subtract:

$\left (\frac{1}{2}x + \frac{2}{3} \right ) - \left (\frac{1}{4}x - \frac{1}{6} \right )$

Possible Answers:

$-\frac{1}{4}x + \frac{5}{6}$

$\frac{1}{4}x + \frac{5}{6}$

$\frac{1}{4}x + \frac{1}{2}$

$\frac{1}{4}x + \frac{1}{9}$

$-\frac{1}{4}x + \frac{1}{2}$

Correct answer:

$\frac{1}{4}x + \frac{5}{6}$

Explanation:

$\left (\frac{1}{2}x + \frac{2}{3} \right ) - \left (\frac{1}{4}x - \frac{1}{6} \right )$

$=\frac{1}{2}x + \frac{2}{3} - \frac{1}{4}x + \frac{1}{6}$

$=\frac{1}{2}x - \frac{1}{4}x + \frac{2}{3} + \frac{1}{6}$

$=\frac{2}{4}x - \frac{1}{4}x + \frac{4}{6} + \frac{1}{6}$

$=\frac{1}{4}x + \frac{5}{6}$

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Example Question #6 : How To Subtract Polynomials

Subtract:

$(7.5x^{2} + 2.4x - 4.6) - (x^{2} - 3.2x - 1.5)$

Possible Answers:

$7.5x^{2} + 5.6x - 3.1$

$6.5x^{2} -0.8x - 6.1$

$6.5x^{2} + 5.6x - 3.1$

$6.5x^{2} -0.8x - 3.1$

$6.5x^{2} + 5.6x - 6.1$

Correct answer:

$6.5x^{2} + 5.6x - 3.1$

Explanation:

$(7.5x^{2} + 2.4x - 4.6) - (x^{2} - 3.2x - 1.5)$

$= 7.5x^{2} + 2.4x - 4.6 - x^{2} + 3.2x +1.5$

$= 7.5x^{2} - 1x^{2} + 2.4x + 3.2x - 4.6 +1.5$

$= 6.5x^{2} + 5.6x - 3.1$

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Example Question #7 : How To Subtract Polynomials

Subtract:

$\left (\frac{3}{4}x + \frac{2}{5} \right ) - \left (\frac{1}{8}x - \frac{1}{10} \right )$

Possible Answers:

$\frac{5}{8}x + \frac{1}{2}$

$\frac{5}{8}x + \frac{1}{15}$

$\frac{5}{8}x - \frac{3}{10}$

$\frac{5}{8}x - \frac{1}{2}$

$\frac{5}{8}x + \frac{3}{10}$

Correct answer:

$\frac{5}{8}x + \frac{1}{2}$

Explanation:

$\left (\frac{3}{4}x + \frac{2}{5} \right ) - \left (\frac{1}{8}x - \frac{1}{10} \right )$

$=\frac{3}{4}x + \frac{2}{5} - \frac{1}{8}x + \frac{1}{10}$

$=\frac{3}{4}x- \frac{1}{8}x + \frac{2}{5} + \frac{1}{10}$

$=\frac{6}{8}x- \frac{1}{8}x + \frac{4}{10} + \frac{1}{10}$

$=\frac{5}{8}x + \frac{5}{10}$

$=\frac{5}{8}x + \frac{1}{2}$

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Example Question #21 : Simplifying Polynomials

Simplify the following expression.

(6x^2-9x+1)-(4x^2+2x-4)

Possible Answers:

x^2+8x-12

10x^2-7x-3

-2x^2+11x-5

2x^2-11x+5

8x^2-7x+3

Correct answer:

2x^2-11x+5

Explanation:

(6x^2-9x+1)-(4x^2+2x-4)

First, we will need to distribute the minus sign.

6x^2-9x+1-4x^2-2x+4)

Then, combine like terms.

(6x^2-4x^2)+(-9x-2x)+(1+4)

2x^2-11x+5

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Example Question #12 : How To Subtract Polynomials

Perform the indicated mathematical operation on these polynomials:

(6x^4+5x^3+12x^2+2x+5)-(6x^4+5x^3+11x^2+2x+5)

Possible Answers:

-x^2

x^2

None of the other answers.

x^2-10

x^2+10

Correct answer:

x^2

Explanation:

We subtract polynomials just like we would anything else, but we must pay attention to the sign after using the distributive property on the sign that separates the two polynomials in subtraction problems. Then, of course, only like terms can combine or subtract from each other.

(6x^4+5x^3+12x^2+2x+5)-(6x^4+5x^3+11x^2+2x+5)

Start by distributing the negative sign into the second polynomial:

6x^4+5x^3+12x^2+2x+5-6x^4-5x^3-11x^2-2x-5)

Now combine or subtract like terms based on the sign between them (same colored terms cancel each other out):

${\color{Red} 6x^4}+{\color{Green} 5x^3}+12x^2+{\color{Blue} 2x}+{\color{Purple} 5}-{\color{Red} 6x^4}-{\color{Green} 5x^3}-11x^2-{\color{Blue} 2x}-{\color{Purple} 5})$

Left over: 12x^2-11x^2=x^2

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Example Question #13 : How To Subtract Polynomials

Subtract the polynomials:

(4x^3+6xy+2y-37)-(6x^2+6xy+y+14)

Possible Answers:

2x^4+3x^2+3x-8

4x^3-6x^2+y-51

4x^3+12x^2+2y+23

4x^3-6x^2+2y-23

4x^3-12x^2+y-51

Correct answer:

4x^3-6x^2+y-51

Explanation:

(4x^3+6xy+2y-37)-(6x^2+6xy+y+14)

Use distribution to simplify the signs of the polynomials.

$4x^3+6{\color{Magenta} xy}+{2\color{Cyan}y}-{{\color{DarkOrange} 37}}-6x^2-6{\color{Magenta} xy}-{\color{Cyan} y}-{\color{DarkOrange} 14}$

Combine like terms while following coefficient rules.

$4x^3+(6-6){{\color{Magenta} xy}}+{(2-1){\color{Cyan} y}}-6x^2-({\color{DarkOrange} {37}-{14}})$

The correct expanded and simplified answer will be

4x^3-6x^2+y-51 .

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Algebra 1 : Polynomial Operations

Study concepts, example questions & explanations for Algebra 1

All Algebra 1 Resources

Example Questions

Example Question #51 : Polynomial Operations

Example Question #1 : How To Subtract Polynomials

Example Question #3 : How To Subtract Polynomials

Example Question #4 : How To Subtract Polynomials

Example Question #5 : How To Subtract Polynomials

Example Question #6 : How To Subtract Polynomials

Example Question #7 : How To Subtract Polynomials

Example Question #21 : Simplifying Polynomials

Example Question #12 : How To Subtract Polynomials

Example Question #13 : How To Subtract Polynomials

All Algebra 1 Resources

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