Pre-Algebra : Multiplying and Dividing Polynomials

Study concepts, example questions & explanations for Pre-Algebra

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Example Questions

Example Question #1 : Multiplying And Dividing Polynomials

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

 

Possible Answers:

\(\displaystyle -2x^5-2x^3+8x\)

\(\displaystyle -2x^3+2x-\frac{2}{x}\)

\(\displaystyle -2x^3-2x+\frac{2}{x}\)

\(\displaystyle -2x^5+2x^3-8x\)

Correct answer:

\(\displaystyle -2x^5+2x^3-8x\)

Explanation:

Multiply making sure to distribute the negative sign:

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

\(\displaystyle =-2x^5+2x^3-8x\)

Example Question #2 : Multiplying And Dividing Polynomials

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

Possible Answers:

\(\displaystyle -x^6+x^3-x^2+6x-2\)

\(\displaystyle -x^5+3x^3-x^2+6x-2\)

\(\displaystyle -x^5+x^3-x^2+6x-2\)

\(\displaystyle -x^6+3x^3-x^2+6x-2\)

Correct answer:

\(\displaystyle -x^5+x^3-x^2+6x-2\)

Explanation:

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

\(\displaystyle =-x^5+x^3-x^2+6x-2\)

Example Question #3 : Multiplying And Dividing Polynomials

Simplify:

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

Possible Answers:

\(\displaystyle 3x^{5}+6x^{4}+4x^{3}-7x^{2}-20\)

\(\displaystyle 3x^{5}+6x^{4}+x^{3}-13x^{2}-5\)

\(\displaystyle 7x^{6}+6x^{4}+8x^{2}-20\)

\(\displaystyle 3x^{5}+6x^{4}+4x^{3}+7x^{2}+20\)

\(\displaystyle 4x^{3}+11x^{2}-16\)

Correct answer:

\(\displaystyle 3x^{5}+6x^{4}+4x^{3}-7x^{2}-20\)

Explanation:

When multiplying, remember the Product Rule of Exponents: \(\displaystyle x^{m}*x^{n}=x^{m+n}\)

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

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

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

\(\displaystyle 4x^{3}+8x^{2}-20\)

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, \(\displaystyle -15x^{2}\) and \(\displaystyle 8x^{2}\) are like terms, but, \(\displaystyle 8x^{2}\) and \(\displaystyle 4x^{3}\) are not like terms:\(\displaystyle 3x^{5}+6x^{4}-15x^{2}+4x^{3}+8x^{2}-20\)

\(\displaystyle 3x^{5}+6x^{4}+4x^{3}-15x^{2}+8x^{2}-20\)

\(\displaystyle 3x^{5}+6x^{4}+4x^{3}-7x^{2}-20\)

 

Example Question #2 : Multiplying And Dividing Polynomials

Expand the following:

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

Possible Answers:

\(\displaystyle \small x^2+5x-14\)

\(\displaystyle \small x^2+9x+14\)

\(\displaystyle \small x^2-5x-14\)

\(\displaystyle \small \small x^2+5x+5\)

\(\displaystyle \small x^2+9x+9\)

Correct answer:

\(\displaystyle \small x^2+5x-14\)

Explanation:

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

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

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

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

\(\displaystyle \small x^2-2x+7x-14\)

\(\displaystyle \small x^2+5x-14\)

Example Question #1 : Multiplying And Dividing Polynomials

Simplify:

\(\displaystyle 4m\cdot3m\)

Possible Answers:

\(\displaystyle 24m\)

\(\displaystyle 12m^{2}\)

\(\displaystyle 12m^{3}\)

\(\displaystyle 12m\)

\(\displaystyle 6m^{2}\)

Correct answer:

\(\displaystyle 12m^{2}\)

Explanation:

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

\(\displaystyle 4m\cdot3m=4\times m\times 3\times m=4\times3\times m\times m\)

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

\(\displaystyle 12\times m\times m\)

Next \(\displaystyle m\) should be multiplied by \(\displaystyle m\), giving us \(\displaystyle m^{2}\).

\(\displaystyle 12\times m^2\)

12 times \(\displaystyle m^{2}\) is equal to \(\displaystyle 12m^{2}\)

Therefore, the correct answer is \(\displaystyle 12m^{2}\).

Example Question #3 : Multiplying And Dividing Polynomials

Multiply:

\(\displaystyle 4x(3x-2)\)

Possible Answers:

\(\displaystyle 12x+8\)

\(\displaystyle 12x^2-8x\)

\(\displaystyle 12x-8\)

\(\displaystyle 12x^2+8x\)

Correct answer:

\(\displaystyle 12x^2-8x\)

Explanation:

Use the distributive property:

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

Example Question #3 : Multiplying And Dividing Polynomials

Multiply:

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

Possible Answers:

\(\displaystyle 1,000Y^{3} - 700Y^{2}+ 490 Y- 343\)

\(\displaystyle 1,000Y^{3} + 343\)

\(\displaystyle 1,000Y^{3} - 100Y^{2}+49 Y- 343\)

\(\displaystyle 1,000Y^{3}+ 100Y^{2}-49 Y- 343\)

\(\displaystyle 1,000Y^{3} + 700Y^{2}- 490 Y- 343\)

Correct answer:

\(\displaystyle 1,000Y^{3} + 343\)

Explanation:

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

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

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

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

So

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

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

Example Question #2 : How To Divide Polynomials

Simplify: \(\displaystyle \frac{6x^7y^3z^9}{3x^6y^3z}\)

 

Possible Answers:

\(\displaystyle 2xz^8\)

\(\displaystyle 6xyz^8\)

\(\displaystyle xyz^8\)

\(\displaystyle xz^8\)

\(\displaystyle xyz\)

Correct answer:

\(\displaystyle 2xz^8\)

Explanation:

Cancel by subtracting the exponents of like terms:

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

Example Question #5 : Multiplying And Dividing Polynomials

Simplify the following expression:

\(\displaystyle (x+4)(x-5)\)

Possible Answers:

=\(\displaystyle x^{2}-20\)

\(\displaystyle =x^{2}+9x+20\)

\(\displaystyle =2x-1\)

\(\displaystyle =-x\)

\(\displaystyle =x^{2}-x-20\)

Correct answer:

\(\displaystyle =x^{2}-x-20\)

Explanation:

Use FOIL (First Outer Inner Last).

\(\displaystyle (x+4)(x-5)\)

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

\(\displaystyle =x^{2}-5x+4x-20\)

\(\displaystyle =x^{2}-x-20\)

Example Question #5 : Multiplying And Dividing Polynomials

Expand the following expression:

\(\displaystyle (x+1)^{3}\)

Possible Answers:

\(\displaystyle =x^{3}+3x^{2}+3x+1\)

\(\displaystyle =3x^{3}+3x^{2}+3x+3\)

\(\displaystyle =x^{3}+1\)

\(\displaystyle =x^{3}+x+1\)

\(\displaystyle =x^{3}+x^{2}+x+1\)

Correct answer:

\(\displaystyle =x^{3}+3x^{2}+3x+1\)

Explanation:

Use FOIL to expand the polynomial.

\(\displaystyle (x+1)^{3}\)

\(\displaystyle =(x+1)(x+1)^{2}\)

\(\displaystyle =(x+1)(x^{2}+2x+1)\)

\(\displaystyle =x^{3}+2x^{2}+x+x^{2}+2x+1\)

\(\displaystyle =x^{3}+3x^{2}+3x+1\)

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