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Pre-Algebra : Multiplying and Dividing Polynomials

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Example Questions

Example Question #31 : Polynomials

Multiply:
-2x(x^4-x^2+4)

Possible Answers:

$-2x^3+2x-\frac{2}{x}$

-2x^5+2x^3-8x

-2x^5-2x^3+8x

$-2x^3-2x+\frac{2}{x}$

Correct answer:

-2x^5+2x^3-8x

Explanation:

Multiply making sure to distribute the negative sign:

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

=-2x^5+2x^3-8x

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Example Question #1 : Multiplying And Dividing Polynomials

Multiply: (x^2+2)(-x^3+3x-1)

Possible Answers:

-x^6+3x^3-x^2+6x-2

-x^5+x^3-x^2+6x-2

-x^5+3x^3-x^2+6x-2

-x^6+x^3-x^2+6x-2

Correct answer:

-x^5+x^3-x^2+6x-2

Explanation:

(x^2+2)(-x^3+3x-1)=-x^5+3x^3-x^2-2x^3+6x-2

=-x^5+x^3-x^2+6x-2

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Example Question #1 : Multiplying And Dividing Polynomials

Simplify:

$\left ( 3x^{2}+4 \right )\left ( x^{3}+2x^{2}-5 \right )$

Possible Answers:

$3x^{5}+6x^{4}+x^{3}-13x^{2}-5$

$3x^{5}+6x^{4}+4x^{3}-7x^{2}-20$

$3x^{5}+6x^{4}+4x^{3}+7x^{2}+20$

$7x^{6}+6x^{4}+8x^{2}-20$

$4x^{3}+11x^{2}-16$

Correct answer:

$3x^{5}+6x^{4}+4x^{3}-7x^{2}-20$

Explanation:

When multiplying, remember the Product Rule of Exponents: $x^{m}*x^{n}=x^{m+n}$

Step 1: Multiply the first term of the first polynomial across the terms of the second polynomial, and then add those products:

$\left ( 3x^{2}*x^{3} \right )+\left ( 3x^{2}*2x^{2} \right )+\left (3x^{2}*-5 \right )$
$3x^{5}+6x^{4}-15x^{2}$ Step 2: Multiply the second term of the first polynomial across the terms of the second polynomial, and again add the products:

$\left ( 4*x^{3} \right )+\left ( 4*2x^{2} \right )+\left (4*-5 \right )$

$4x^{3}+8x^{2}-20$

Step 3: Add the products from Step 1 and Step 2 by combining like terms. Remember that variables with different exponents are not like terms. For example, $-15x^{2}$ and $8x^{2}$ are like terms, but, $8x^{2}$ and $4x^{3}$ are not like terms: $3x^{5}+6x^{4}-15x^{2}+4x^{3}+8x^{2}-20$

$3x^{5}+6x^{4}+4x^{3}-15x^{2}+8x^{2}-20$

$3x^{5}+6x^{4}+4x^{3}-7x^{2}-20$

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Example Question #2 : Multiplying And Dividing Polynomials

Expand the following:

$\small (x+7)(x-2)$

Possible Answers:

$\small x^2+5x-14$

$\small x^2-5x-14$

$\small x^2+9x+14$

$\small \small x^2+5x+5$

$\small x^2+9x+9$

Correct answer:

$\small x^2+5x-14$

Explanation:

Recall that when expanding polynomials, we use the term FOIL (First, Outside, Inside, Last) to help us multiply all terms together.

$\small (x+7)(x-2)$

$\small \small (x)(x)+(x)(-2)+(7)(x)+(7)(-2)$

Next, multiply each term to simplify and combine like terms. Note: Be careful with negative signs.

$\small x^2-2x+7x-14$

$\small x^2+5x-14$

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Example Question #6 : Variables

Simplify:

$4m\cdot3m$

Possible Answers:

12m

$12m^{2}$

$12m^{3}$

24m

$6m^{2}$

Correct answer:

$12m^{2}$

Explanation:

To solve, you can use the commutative and associative properties of multiplication to group like-terms together.

$4m\cdot3m=4\times m\times 3\times m=4\times3\times m\times m$

The 4 and 3 should be first multiplied, resulting in 12.

$12\times m\times m$

Next should be multiplied by , giving us $m^{2}$ .

$12\times m^2$

12 times $m^{2}$ is equal to $12m^{2}$ .

Therefore, the correct answer is $12m^{2}$ .

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Example Question #1 : Multiplying And Dividing Polynomials

Multiply:

4x(3x-2)

Possible Answers:

12x^2+8x

12x^2-8x

12x-8

12x+8

Correct answer:

12x^2-8x

Explanation:

Use the distributive property:

$4x(3x-2)=(4x\times3x)-(4x\times2)=12x^2-8x$

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Example Question #1 : Multiplying And Dividing Polynomials

Multiply:

$\left (100Y^{2}- 70Y + 49 \right )(10Y+7)$

Possible Answers:

$1,000Y^{3} + 343$

$1,000Y^{3} + 700Y^{2}- 490 Y- 343$

$1,000Y^{3}+ 100Y^{2}-49 Y- 343$

$1,000Y^{3} - 700Y^{2}+ 490 Y- 343$

$1,000Y^{3} - 100Y^{2}+49 Y- 343$

Correct answer:

$1,000Y^{3} + 343$

Explanation:

$\left (100Y^{2}- 70Y + 49 \right )(10Y+7)$

$= \left [\left (10Y \right )^{2}- 10Y\cdot 7 +7^{2} \right ](10Y+7)$

This product fits the sum of cubes pattern, where A = 10Y, B = 7 :

$(A^{2}-AB +B^{2})(A+B) = A^{3}+B^{3}$

$\left [\left (10Y \right )^{2}- 10Y\cdot 7 +7^{2} \right ](10Y+7)$

$= (10Y)^{3}+7^{3} = 1,000Y^{3} + 343$

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Example Question #41 : Polynomials

Simplify: $\frac{6x^7y^3z^9}{3x^6y^3z}$

Possible Answers:

2xz^8

xyz

6xyz^8

xyz^8

xz^8

Correct answer:

2xz^8

Explanation:

Cancel by subtracting the exponents of like terms:

$\frac{6x^7y^3z^9}{3x^6y^3z} = 2x^{7-6}y^{3-3}z^{9-1}=2xy^0z^8=2xz^8$

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Example Question #3 : Multiplying And Dividing Polynomials

Simplify the following expression:

(x+4)(x-5)

Possible Answers:

=2x-1

= $x^{2}-20$

=-x

$=x^{2}-x-20$

$=x^{2}+9x+20$

Correct answer:

$=x^{2}-x-20$

Explanation:

Use FOIL (First Outer Inner Last).

(x+4)(x-5)

=x(x-5)+4(x-5)

$=x^{2}-5x+4x-20$

$=x^{2}-x-20$

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Example Question #5 : Multiplying And Dividing Polynomials

Expand the following expression:

$(x+1)^{3}$

Possible Answers:

$=x^{3}+3x^{2}+3x+1$

$=3x^{3}+3x^{2}+3x+3$

$=x^{3}+1$

$=x^{3}+x+1$

$=x^{3}+x^{2}+x+1$

Correct answer:

$=x^{3}+3x^{2}+3x+1$

Explanation:

Use FOIL to expand the polynomial.

$(x+1)^{3}$

$=(x+1)(x+1)^{2}$

$=(x+1)(x^{2}+2x+1)$

$=x^{3}+2x^{2}+x+x^{2}+2x+1$

$=x^{3}+3x^{2}+3x+1$

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Pre-Algebra : Multiplying and Dividing Polynomials

Study concepts, example questions & explanations for Pre-Algebra

All Pre-Algebra Resources

Example Questions

Example Question #31 : Polynomials

Example Question #1 : Multiplying And Dividing Polynomials

Example Question #1 : Multiplying And Dividing Polynomials

Example Question #2 : Multiplying And Dividing Polynomials

Example Question #6 : Variables

Example Question #1 : Multiplying And Dividing Polynomials

Example Question #1 : Multiplying And Dividing Polynomials

Example Question #41 : Polynomials

Example Question #3 : Multiplying And Dividing Polynomials

Example Question #5 : Multiplying And Dividing Polynomials

All Pre-Algebra Resources

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