Question

How do you find the product $5w( - 3{w^2} + 2w - 4)$?

Answer 1

Hint: In this question, we want to find the product of the given expression. Here, one term is monomial and the other term is polynomial. The multiplication of polynomials by monomial means every term of the polynomial is multiplied by the monomial. That means we are applying the distributive property of multiplication. The distributive property states that when a term is multiplied outside parentheses, we multiply it to every term inside the parentheses. The general formula for the distributive property is $a\left( {b + c} \right) = ab + ac$.

Complete step-by-step solution:
In this question, we want to find the product of $5w( - 3{w^2} + 2w - 4)$.
Here, $5w$ is monomial and $( - 3{w^2} + 2w - 4)$is a polynomial.
$ \Rightarrow 5w( - 3{w^2} + 2w - 4)$
Let us apply the distributive property to the above expression.
Here, a term $5w$ is multiplied outside parentheses. So, we will multiply it to every term of $( - 3{w^2} + 2w - 4)$.
Therefore,
$ \Rightarrow 5w( - 3{w^2}) + 5w\left( {2w} \right) + 5w\left( { - 4} \right)$
Let us apply the multiplication. For that, we will directly multiply the coefficients and then combine the like terms. To solve the multiplication of like terms, we will add the exponents of the same base.
 $ \Rightarrow \left( { - 15 \times w \times {w^2}} \right) + \left( {10 \times w \times w} \right) + \left( { - 20 \times w} \right)$
As we know the formula of the multiplication of exponents is${x^a} \times {x^b} = {x^{a + b}}$.
Therefore,
 $ \Rightarrow \left( { - 15 \times {w^{1 + 2}}} \right) + \left( {10 \times {w^{1 + 1}}} \right) + \left( { - 20 \times w} \right)$
Let us apply the addition to the exponents. That is equal to,
 $ \Rightarrow - 15{w^3} + 10{w^2} - 20w$

Hence, the multiplication of a given expression is $ - 15{w^3} + 10{w^2} - 20w$.

Note: If we want to verify our answer, we simply take out the common factors among all the terms. After taking out the common factor we can get the expression that is given in the question.
In this question, we get the answer
 $ \Rightarrow - 15{w^3} + 10{w^2} - 20w$
Here, the common factor from the coefficients is 5 and the common factor from the variable is w.
That is equal to,
$ \Rightarrow 5w( - 3{w^2} + 2w - 4)$
Hence, we get the expression that is given in the question.