complete the sentence below. if (3x² + 22x + 7)÷(x + 7)=3x + 1, then (x + 7)( )=( ). the check of the…

complete the sentence below. if (3x² + 22x + 7)÷(x + 7)=3x + 1, then (x + 7)( )=( ). the check of the polynomial division problem shows that the product of two polynomials is a polynomial. this supports the fact that the ( ) property is satisfied for polynomial multiplication.

complete the sentence below. if (3x² + 22x + 7)÷(x + 7)=3x + 1, then (x + 7)( )=( ). the check of the polynomial division problem shows that the product of two polynomials is a polynomial. this supports the fact that the ( ) property is satisfied for polynomial multiplication.

Answer

Explanation:

Step1: Recall division - multiplication relationship

If $\frac{a}{b}=c$, then $b\times c = a$. Given $\frac{3x^{2}+22x + 7}{x + 7}=3x + 1$, so $(x + 7)(3x+1)=3x^{2}+22x + 7$.

Step2: Identify the property

The fact that the product of two polynomials $(x + 7)$ and $(3x + 1)$ is a polynomial $3x^{2}+22x + 7$ shows that the closure - property is satisfied for polynomial multiplication. The closure property of multiplication for a set of polynomials states that when you multiply two polynomials from the set, the result is also a polynomial in the set.

Answer:

First blank: $3x + 1$ Second blank: $3x^{2}+22x + 7$ Third blank: closure