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Lesson Example Discussion Quiz: Class Homework

Step-1

Title: Multiplication of Polynomials

Grade: 8-a Lesson: S1-L2

Explanation: Hello Students, time to practice and review the steps for the problem.

Lesson Steps

Step	Type	Explanation	Answer
1	Problem	Find the product of the following polynomials: \$(4x^3 − 3x^2+ 2x −1)(x^4−2x^3 + 3x^2− 4x + 5)\$
2	Step	The given polynomials expression is	\$(4x^3 − 3x^2+ 2x −1)(x^4−2x^3 + 3x^2− 4x + 5)\$
3	Step	Now, let’s first compute the \$4x^3\$ multiplication	\$4x^3(x^4 − 2x^3 + 3x^2 − 4x + 5)\$ \$= 4x^7 - 8x^6 + 12x^5 - 16x^4 +20x^3\$
4	Step	Then compute the \$- 3x^2\$ multiplication	\$-3x^2 (x^4 −2x^3 + 3x^2 − 4x + 5)\$ \$= -3x^6 + 6x^5 - 9x^4 + 12x^3 -15x^2\$
5	Step	Next compute the 2x multiplication	\$2x (x^4 − 2x^3 + 3x^2 − 4x + 5)\$ \$= 2x^5 - 4x^4 + 6x^3 - 8x^2 + 10x\$
6	Step	Then compute the -1 multiplication	\$-1(x^4−2x^3+3x^2−4x+5)\$ \$= -1x^4 + 2x^3 - 3x^2 + 4x -5\$
7	Step	Now, sum up all the results:	\$4x^7 - 8x^6 + 12x^5 - 16x^4 +20x^3 - 3x^6 + 6x^5 - 9x^4 + 12x^3 -15x^2 + 2x^5 - 4x^4 + 6x^3 - 8x^2 + 10x -1x^4 + 2x^3 - 3x^2 + 4x -5\$
8	Step	Make it simplify the expression	\$4x^7 -11x^6 +20x^5 -30x^4 + 40x^3 -26x^2 + 14x -5\$
9	Step	So, the product of the given polynomials is \$4x^7 -11x^6 +20x^5 -30x^4 + 40x^3 -26x^2 + 14x -5\$.
10	Choice.A	Correct: It performed the accurate calculation	\$4x^7 -11x^6 +20x^5 -30x^4 + 40x^3 -26x^2 + 14x -5\$
11	Choice.B	Wrong due to different signs for some terms, so it’s incorrect	\$4x^7 - 11x^6 - 20x^5 - 27x^4 + 40x^3 -26x^2 +14x -5\$
12	Choice.C	It is wrong because it has a different combination of coefficients for the terms	\$4x^7 -11x^6 +20x^5 -30x^4 - 40x^3 + 26x^2 + 14x -5\$
13	Choice.D	This is inaccurate due to incorrect signs across multiple terms	\$4x^7 + 11x^6 - 20x^5 + 27x^4 + 40x^3 + 26x^2 +14x + 5\$
14	Answer	Option	A
15	Sumup	Can you summarize what you’ve understood in the above steps?

Step

Type

Explanation

Answer

Problem

Find the product of the following polynomials: \$(4x^3 − 3x^2+ 2x −1)(x^4−2x^3 + 3x^2− 4x + 5)\$

Step

The given polynomials expression is

\$(4x^3 − 3x^2+ 2x −1)(x^4−2x^3 + 3x^2− 4x + 5)\$

Step

Now, let’s first compute the \$4x^3\$ multiplication

\$4x^3(x^4 − 2x^3 + 3x^2 − 4x + 5)\$

\$= 4x^7 - 8x^6 + 12x^5 - 16x^4 +20x^3\$

Step

Then compute the \$- 3x^2\$ multiplication

\$-3x^2 (x^4 −2x^3 + 3x^2 − 4x + 5)\$

\$= -3x^6 + 6x^5 - 9x^4 + 12x^3 -15x^2\$

Step

Next compute the 2x multiplication

\$2x (x^4 − 2x^3 + 3x^2 − 4x + 5)\$

\$= 2x^5 - 4x^4 + 6x^3 - 8x^2 + 10x\$

Step

Then compute the -1 multiplication

\$-1(x^4−2x^3+3x^2−4x+5)\$

\$= -1x^4 + 2x^3 - 3x^2 + 4x -5\$

Step

Now, sum up all the results:

\$4x^7 - 8x^6 + 12x^5 - 16x^4 +20x^3 - 3x^6 + 6x^5 - 9x^4 + 12x^3 -15x^2 + 2x^5 - 4x^4 + 6x^3 - 8x^2 + 10x -1x^4 + 2x^3 - 3x^2 + 4x -5\$

Step

Make it simplify the expression

\$4x^7 -11x^6 +20x^5 -30x^4 + 40x^3 -26x^2 + 14x -5\$

Step

So, the product of the given polynomials is \$4x^7 -11x^6 +20x^5 -30x^4 + 40x^3 -26x^2 + 14x -5\$.

Choice.A

Correct: It performed the accurate calculation

\$4x^7 -11x^6 +20x^5 -30x^4 + 40x^3 -26x^2 + 14x -5\$

Choice.B

Wrong due to different signs for some terms, so it’s incorrect

\$4x^7 - 11x^6 - 20x^5 - 27x^4 + 40x^3 -26x^2 +14x -5\$

Choice.C

It is wrong because it has a different combination of coefficients for the terms

\$4x^7 -11x^6 +20x^5 -30x^4 - 40x^3 + 26x^2 + 14x -5\$

Choice.D

This is inaccurate due to incorrect signs across multiple terms

\$4x^7 + 11x^6 - 20x^5 + 27x^4 + 40x^3 + 26x^2 +14x + 5\$

Answer

Option

Sumup

Can you summarize what you’ve understood in the above steps?