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Chapter 4: Problem 38

Find each product. $$ (2 y+8)\left(3 y^{4}-2 y^{2}+1\right) $$

Short Answer

Expert verified
The product is \(6y^5 + 24y^4 - 4y^3 - 16y^2 + 2y + 8\).

Step by step solution

01

Apply the Distributive Property

First, use the distributive property (also known as the FOIL method for binomials) to multiply each term in the first binomial by each term in the second binomial. Write it out as: \( (2y + 8) \times (3y^4 - 2y^2 + 1) = 2y \times (3y^4 - 2y^2 + 1) + 8 \times (3y^4 - 2y^2 + 1) \)
02

Distribute the First Term

Now, distribute the first term in the first binomial: \(2y \times 3y^4 = 6y^5 2y \times (-2y^2) = -4y^3 2y \times 1 = 2y \)Combine these results to get: \(6y^5 - 4y^3 + 2y \).
03

Distribute the Second Term

Next, distribute the second term in the first binomial: \(8 \times 3y^4 = 24y^4 8 \times (-2y^2) = -16y^2 8 \times 1 = 8 \)Combine these results to get: \(24y^4 - 16y^2 + 8 \).
04

Combine Like Terms

Now, combine all the terms from Step 2 and Step 3:\(6y^5 - 4y^3 + 2y + 24y^4 - 16y^2 + 8\). Arrange them in descending powers of y to get the final simplified form: \(6y^5 + 24y^4 - 4y^3 - 16y^2 + 2y + 8\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

headline of the respective core concept
Polynomial multiplication involves expanding and simplifying expressions where each term in one polynomial is multiplied by each term in another polynomial. In this exercise, you multiply (2y + 8) by (3y^4 - 2y^2 + 1).
  • First, apply the distributive property.
  • Each term in the first polynomial multiplies each term in the second polynomial.
  • This process can be considered like a double-layer FOIL method.
By properly doing the calculations, you ensure the polynomial multiplication is accurate.
headline of the respective core concept
Combining like terms is crucial when simplifying polynomials. Like terms have the same variable raised to the same power. After multiplying all terms, group them to simplify the expression.
  • For example, terms with y^5, y^4, etc., should be combined separately.
  • It makes the final polynomial cleaner and easier to interpret.
From the exercise: combine 6y^5, 24y^4, -4y^3, -16y^2, 2y, and 8.This results in the simplified expression: 6y^5 + 24y^4 - 4y^3 - 16y^2 + 2y + 8.
headline of the respective core concept
Understanding algebraic expressions is fundamental in math. They involve variables and constants combined using arithmetic operations. The expression (2y + 8) and(3y^4 - 2y^2 + 1) are algebraic.
  • These expressions can be simplified and solved using rules for polynomial multiplication.
  • Recognizing and manipulating these expressions is key in algebra.
Practicing problems like this helps in mastering the use of different algebraic operations efficiently.

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Most popular questions from this chapter

Find each product. Recall that \(a^{2}=a \cdot a\) and \(a^{3}=a^{2} \cdot a\). $$ (m-5)^{3} $$

Find each product. Recall that \(a^{2}=a \cdot a\) and \(a^{3}=a^{2} \cdot a\). $$ (2 p-5)^{2} $$

\(\frac{\left(9^{-1} z^{-2} x\right)^{-1}\left(4 z^{2} x^{4}\right)^{-2}}{\left(5 z^{-2} x^{-3}\right)^{2}}\)

Multiply each of the following. (a) \((x-1)\left(x^{2}+x+1\right)\) (b) \((y-2)\left(y^{2}+2 y+4\right)\) Describe the final product as "the ___________of two_______."

Use scientific notation to calculate the result in each expression. Write answers in scientific notation. 73\. \(\frac{650,000,000(0.0000032)}{0.00002}\)
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