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Chapter 4: Problem 40

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

Short Answer

Expert verified
The product is \(2a^5 + a^4 - a^3 + a^2 - a + 3\).

Step by step solution

01

Distribute the First Term

Multiply the first term in the binomial, which is \(2a\), by each term in the polynomial \(a^4 - a^3 + a^2 - a + 1\): \[(2a) \times a^4 = 2a^5\] \[(2a) \times (-a^3) = -2a^4\] \[(2a) \times a^2 = 2a^3\] \[(2a) \times (-a) = -2a^2\] \[(2a) \times 1 = 2a\]
02

Distribute the Second Term

Now multiply the second term in the binomial, which is \(3\), by each term in the polynomial \(a^4 - a^3 + a^2 - a + 1\): \[3 \times a^4 = 3a^4\] \[3 \times (-a^3) = -3a^3\] \[3 \times a^2 = 3a^2\] \[3 \times (-a) = -3a\] \[3 \times 1 = 3\]
03

Add the Results

Add the results from Step 1 and Step 2 together: \[2a^5 - 2a^4 + 2a^3 - 2a^2 + 2a\] \[+ 3a^4 - 3a^3 + 3a^2 - 3a + 3\] Combine like terms:\[2a^5 + ( -2a^4 + 3a^4) + (2a^3 - 3a^3) + (-2a^2 + 3a^2) + (2a - 3a) + 3 = 2a^5 + a^4 - a^3 + a^2 - a + 3\]

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Key Concepts

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

Binomial Distribution
Let's start with understanding binomial distribution in the context of polynomial multiplication. When we have a binomial, which is an expression with two terms (like \(2a + 3\)), and we multiply it by a polynomial (like \(a^4 - a^3 + a^2 - a + 1\)), we distribute each term of the binomial across every term of the polynomial.

This means that for each term in the binomial, you will multiply it by each term in the polynomial individually.
For example:
  • Multiply \(2a\) by each term in the polynomial.
  • Multiply \(3\) by each term in the polynomial.

Then, combine all these individual products together. This systematic approach ensures every term is accounted for without missing any possible product.
Combining Like Terms
After distributing and multiplying, you'll get a longer polynomial that often contains 'like terms'. These are terms that have the same variable raised to the same power.

For instance, in our example, the terms \( -2a^4\) and \(3a^4\) are like terms because they both have \(a^4\).

Combining like terms means to:
  • Add or subtract the coefficients of these terms.
  • This process reduces the polynomial to its simplest form.

In practical terms, you take each set of like terms from your distributions and perform the addition or subtraction.
As an example: \[ -2a^4 + 3a^4 = a^4 \] This simplification makes the polynomial easier to read and work with.
Algebraic Expressions
Algebraic expressions involve variables, constants, and operators. In our task, we're given expressions that contain variables raised to different powers.

To handle these expressions, we need some key operations:
  • Multiplication of terms (as seen in the distributive steps).
  • Combining like terms (to simplify our result).

Algebraic expressions can represent real-world situations and are foundational in algebra.
When working with algebraic expressions:
  • Always keep track of the coefficients (numbers in front of variables) and exponents (powers of the variables).
  • Ensure your final expression is as simplified as possible by combining like terms.

This methodical approach keeps your work organized and increases the accuracy of your solutions.

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Most popular questions from this chapter

Add or subtract as indicated. \((4 x+2 x y-3)-(-2 x+3 x y+4)\)

Find each product. $$ (2 m-3 n)(m+5 n) $$

Add or subtract as indicated. \(\left(3 k^{2} h^{3}+5 k h+6 k^{3} h^{2}\right)-\left(2 k^{2} h^{3}-9 k h+k^{3} h^{2}\right)\)

Find each product. Recall that \(a^{2}=a \cdot a\) and \(a^{3}=a^{2} \cdot a\). $$ -4 r(3 r+2)(2 r-5) $$

The special product \((x+y)(x-y)=x^{2}-y^{2}\) can be used to perform some multiplications.Example: $$\begin{array}{l|l}51 \times 49 & 102 \times 98 \\\=(50+1)(50-1) & =(100+2)(100-2) \\\=50^{2}-1^{2} & =100^{2}-2^{2} \\\=2500-1 & =10,000-4 \\\=2499 & =9996\end{array}$$ Use this method to calculate each product mentally. $$ 30 \frac{1}{3} \times 29 \frac{2}{3} $$
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