Question

1. higher order thinking is it possible for the product of a monomial and trinomial to be a binomial? explain.
2. mathematical connections a triangle has a height of 2x + 6 and a base length of x + 4. what is the area of the triangle?
3. communicate precisely explain how to find the combined volume of the two rectangular prisms described. one has side lengths of 3x, 2x + 1, and x + 3. the other has side lengths of 5x - 2, x + 9, and 8.

Explanation:

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Step1: Recall polynomial multiplication

A monomial is of the form $a$, and a trinomial is of the form $b + c + d$. Their product is $a(b + c + d)=ab+ac + ad$. But if one of the terms in the trinomial is 0, for example, if the trinomial is $b+0 + d$ and the monomial is $a$, then $a(b + 0 + d)=ab+ad$, which is a binomial.

Step1: Recall triangle - area formula

The area formula of a triangle is $A=\frac{1}{2}bh$, where $b$ is the base and $h$ is the height. Here, $b=x + 4$ and $h=2x + 6$.

Step2: Substitute values into formula

$A=\frac{1}{2}(x + 4)(2x + 6)$.

Step3: Expand the product

First, expand $(x + 4)(2x + 6)=x(2x+6)+4(2x + 6)=2x^{2}+6x+8x + 24=2x^{2}+14x + 24$. Then $A=\frac{1}{2}(2x^{2}+14x + 24)=x^{2}+7x + 12$.

Step1: Recall rectangular - prism volume formula

The volume formula of a rectangular prism is $V = lwh$, where $l$, $w$, and $h$ are the side - lengths.
For the first prism with side - lengths $3x$, $2x + 1$, and $x + 3$, its volume $V_1=3x(2x + 1)(x + 3)$.
Expand $(2x + 1)(x + 3)=2x^{2}+6x+x + 3=2x^{2}+7x + 3$. Then $V_1=3x(2x^{2}+7x + 3)=6x^{3}+21x^{2}+9x$.
For the second prism with side - lengths $5x-2$, $x + 9$, and 8, its volume $V_2=8(5x-2)(x + 9)$.
Expand $(5x-2)(x + 9)=5x^{2}+45x-2x-18=5x^{2}+43x-18$. Then $V_2=8(5x^{2}+43x-18)=40x^{2}+344x-144$.

Step2: Find the combined volume

The combined volume $V = V_1+V_2=(6x^{3}+21x^{2}+9x)+(40x^{2}+344x-144)=6x^{3}+(21x^{2}+40x^{2})+(9x + 344x)-144=6x^{3}+61x^{2}+353x-144$.
To find the combined volume, first find the volume of each rectangular prism using the formula $V = lwh$, expand the products for each volume, and then add the two volumes together.

Answer:

Yes, it is possible. If one of the terms in the trinomial is zero, the product of a monomial and a trinomial can be a binomial.

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