Question

Good evening trying to help my son with home work an were stuck on this problem pleased help. 1.What is the product of (2x – 4) and (5x^2-2x+6)? Write your answer in standard form.(a)Show your work.(b)Is the product of (2x – 4) and (5x^2+2x+6) equal to the product of (4 – 2x) and (5x^2-2x+6 )? Explain your answer.

Answer 1

`\begin{gathered} a)\text{ }10x^3\text{ }-24x^2\text{ + 20x - 24} \\ b)\text{ the products are not equal} \end{gathered}`

Step-by-step explanation:

1a) Product means multiplication. So we'll be multiplying (2x - 4) by (5x² - 2x + 6)

Using distributive property:

`\begin{gathered} \mleft(2x-4\mright)*(5x^2-2x+6) \\ \text{Multiply 2x }by\text{ }(5x^2-2x+6)\text{ and -4 by }(5x^2-2x+6)\colon \\ =\text{ 2x}(5x^2-2x+6)-4(5x^2-2x+6)_{} \\ \\ Use\text{ the term outside to multiply each of the inner terms:} \\ =2x(5x^2)-2x(2x)+2x(6)-4(5x^2)\text{ -4(-2x) -4(+6)} \end{gathered}`

Simplify:

`\begin{gathered} To\text{ expand the parenthesis, we n}ed\text{ to consider the signs:} \\ $$mu\text{ltiplication of same signs give positive sign}$$ \\ $$m\text{ ultiplication of opposite signs give negative sign}$$. \\ 2x\mleft(5x^2\mright)=10x^3 \\ -2x(+2x)\text{ = }-4x^2 \\ 2x(6)\text{ = 12x} \\ -4(+5x^2)=-20x^2 \\ -4(-2x)\text{ = 8x} \\ -4(+6)\text{ = -24} \end{gathered}`
`\begin{gathered} 2x(5x^2)-2x(2x)+2x(6)-4(5x^2)\text{ -4(-2x) -4(+6)} \\ =10x^3-4x^2\text{ + 12x }-20x^2\text{ + 8x - 24} \end{gathered}`

Collect like terms:

`\begin{gathered} \text{Bring terms that are alike together- terms that have same exponent of x together} \\ =10x^3-4x^2\text{ }-20x^2\text{ + 12x + 8x - 24} \\ =\text{ }10x^3\text{ }-24x^2\text{ + 20x - 24} \\ \\ \text{Hence, product of (2x - 4) and (5x}^2\text{ - 2x + 6) = }10x^3\text{ }-24x^2\text{ + 20x - 24} \end{gathered}`

Standard form of a polynomial is when the exponents are written in descending order.

From our result above, the exponents are decreasing from left to right

1b) to determine if the product of (2x – 4) and (5x²+2x+6) is equal to the product of (4 – 2x) and (5x²-2x+6), we will do the product of the second and compare with the product we got.

Without calculating, we can also tell if they are equal or not. We will just compare the expressions from both product

2x - 4 is not the same as 4 - 2x

Also 5x²+2x+6 is not the same as 5x²-2x+6

The signs make them different

As a result, the result of the expansion will be different

Expanding to ascertain:

`\begin{gathered} (4-2x)(5x^2\text{ - 2x + 6)} \\ =\text{ 4(}5x^2\text{ - 2x + 6) - 2x(}5x^2\text{ - 2x + 6)} \\ =20x^2-8x+24-10x^{3\text{ }}+4x^2\text{ - 12}x \\ =-10x^3\text{ }+20x^2+4x^2\text{ -8x - 12x +24} \\ (4-2x)(5x^2\text{ - 2x + 6) }=\text{ }-10x^3\text{ }+24x^2\text{ - 20x +24} \end{gathered}`

We see the result from both products are different. This is due to the signs

Hence, product of (2x – 4) and (5x^2+2x+6) is not equal to the product of (4 – 2x) and (5x^2-2x+6)