How to multiply a monomial by a polynomial - Algebra

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Question

Distribute:

$4x(2x^{2}+3x-4)$

Answer

Be sure to distribute the along with its coefficient.

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Question

Answer

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Question

Answer

$= 3 x^4\cdot 2 x^2 + 3 x^4 (-4 x)+3 x^4\cdot 3$

$= (3\times 2)x^4×x^2 + (3\times -4)x^4 x+(3\times 3)x^4$

$= 6x^{4+2}+ (-12)x^{4+1}+9x^4$

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Question

Expand:

Answer

To expand, multiply 8x by both terms in the expression (3x + 7).

8x multiplied by 3x is 24x².

8x multiplied by 7 is 56x.

Therefore, 8x(3x + 7) = 24x² + 56x.

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Question

Write as a polynomial.

Answer

We need to distribute the 4x2 through the terms in the parentheses:

This becomes .

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Question

Find the product:

$\small 7n(8n-2)$

Answer

First, mulitply the mononomial by the first term of the polynomial:

$\small 7n\times8n\ = 56n^2$

Second, multiply the monomial by the second term of the polynomial:

$\small 7n\times (-2)\ = -14n$

Add the terms together:

$\small 56n^2\ +\ (-14n)\ = 56n^2-14n$

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Question

Find the product: $5g(g^{2} - 4)$

Answer

times $g^{2}$ gives us $5g^{3}$ , while times 4 gives us . So it equals $5g^{3}-20g$ .

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Question

$4x^{2}y\left ( 2xy + 6xy^{2} + 3y^{{3}}\right ) =$

Answer

Use the distributive property to obtain each term:

$4x^{2}y\left ( 2xy + 6xy^{2} + 3y^{{3}}\right ) =$

$4x^{2}y(2xy) + 4x^{2}y(6xy^{2}) + 4x^{2}y(3y^{3}) =$

$8x^{3}y^{2} + 24x^{3}y^{3} + 12x^{2}y^{4}$

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Question

Simplify the following expression.

$4x(3x^{2}+x+10)$

Answer

Distribute the outside term into the parentheses.

$4x(3x^{2}+x+10)=(4x3x^{2})+(4x x) +(4x* 10)$

Simplify each distributed factor into one expression.

$12x^{3}+4x^{2}+40x$

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Question

Simplify the following expression.

$5y(3y^{3}-2y+9)$

Answer

Distribute the outside term into the parentheses.

$5y(3y^{3}-2y+9)=(5y* 3y^{3})- (5y* 2y) + (5y*9)$

Simplify each distributed factor into one expression.

$15y^{4}-10y^{2}+45y$

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Question

$\textup{What is the product of }2x\textup{ and }3x^{2}-5x+2\textup{?}$

Answer

$\textup{Distribute: Multiply each term in the polynomial by }2x:$

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

$6x^{3}-10x^{2}+4x$

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Question

Evaluate.

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

Answer

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

Using the distributive property you are simply going to share the term , with every term in the poynomial

$ab\cdot ab^{2}+ ab\cdot 4a- ab\cdot b$

Now because we are multiplying like variables we can add the exponents, to simplify each expression

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

This will be our final answer because we can not add terms unless they are 'like' meaning they contain the same elements and degrees.

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Question

Simplify the following expression.

$3y (-y^{2}-18y+7)$

Answer

Distribute and multiply by each of the terms within the parentheses.

$3y \times -y^{2}=-3y^{3}$

$3y\times-18y=-54y^{2}$

$3y\times7=21y$

Regroup the resulting terms

$-3y^{3}-54y^{2}+21y$

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Question

Simplify the following expression.

Answer

Distribute to each of the terms within the parentheses.

$3x\times-4x=-12x^{2}$

$3x\times7=21x$

Putting it back together...

$-12x^{2}+21x$

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Question

Simplify the following expression.

$-2x^{2}(2x^{2}-11x)$

Answer

$-2x^{2}(2x^{2}-11x)$

Distribute $-2x^{2}$ to each term within parentheses.

$-2x^{2}(2x^{2})=-4x^{4}$

$-2x^{2}(-11x)=22x^{3}$

Putting it back together...

$-4x^{4}+22x^{3}$

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Question

Expand the expression by multiplying the terms.

$\small (x-4)(x+2)(2x-5)$

Answer

$\small (x-4)(x+2)(2x-5)$

When multiplying, the order in which you multiply does not matter. Let's start with the first two monomials.

Use FOIL to expand.

Now we need to multiply the third monomial.

$\small (x-4)(x+2)(2x-5)=(x^2-2x-8)(2x-5)$

Similar to FOIL, we need to multiply each combination of terms.

Combine like terms.

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Question

Multiply:

$-3t ^{2} ( 2t ^{2} - 5t + 7)$

Answer

$-3t ^{2} ( 2t ^{2} - 5t + 7)$

$= -3t ^{2} \cdot 2t ^{2} - \left (-3t ^{2} \right )\cdot 5t +\left (-3t ^{2} \right )\cdot 7$

$= -3t ^{2} \cdot 2t ^{2} + 3t ^{2} \cdot 5t -3t ^{2}\cdot 7$

$= -3\cdot 2 \cdot t ^{2} \cdot t ^{2} + 3\cdot 5 \cdot t ^{2} \cdot t -3\cdot 7 \cdot t ^{2}$

$= -6 \cdot t ^{2+2} + 15\cdot t ^{2+1} -21 \cdot t ^{2}$

$= -6 t ^{4} + 15t ^{3} -21t ^{2}$

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Question

Simplify the following:

$(2p^{2})(4x+p)$

Answer

Distribute the $2p^{2}$ to each term in the parentheses in the other polynomial.

$2p^{2}(4x)=8xp^{2}$

$2p^{2}(p)=2p^{3}$

Putting the results back together

$(2p^{2})(4x+p)=8xp^{2}+2p^{3}$

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Question

Simplify the following

$(7y)(7t-3y^{3} + 2)$

Answer

Distribute to each term in the parentheses in the polynomial

$7y(-3y^{3})=-21y^{4}$

Combine the results

$(7y)(7t-3y^{3} + 2)=49yt-21y^{4} +14y$

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Question

Multiply:

Answer

Multiply each term of the polynomial by . Be careful to distribute the negative sign.

Add the individual terms together:

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