Problem 49 Multiply the polynomials. $$(a... [FREE SOLUTION]

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Prealgebra

Julie Miller, Molly O鈥橬eill, Nancy Hyde

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Chapter 10: Problem 49

Multiply the polynomials. $$(a+5)\left(a^{2}-2 a+4\right)$$

Short Answer

Expert verified

The result is $a^3 + 3a^2 - 6a + 20$.

Step by step solution

Distribute the first term

Multiply the first term of the first polynomial, which is $a$, by each term in the second polynomial: \[a \times (a^2 - 2a + 4)\]This gives: \[a \times a^2 = a^3\]\[a \times (-2a) = -2a^2\]\[a \times 4 = 4a\]

Distribute the second term

Multiply the second term of the first polynomial, which is $5$, by each term in the second polynomial: \[5 \times (a^2 - 2a + 4)\]This gives: \[5 \times a^2 = 5a^2\]\[5 \times (-2a) = -10a\]\[5 \times 4 = 20\]

Combine like terms

Now, combine all the products obtained in steps 1 and 2 together: \[a^3 - 2a^2 + 4a + 5a^2 - 10a + 20\]Combine the like terms: \[a^3 + ( -2a^2 + 5a^2) + (4a - 10a) + 20\]This simplifies to: \[a^3 + 3a^2 - 6a + 20\]

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Key Concepts

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

Distributive Property

The distributive property is essential when multiplying polynomials. It allows us to distribute the multiplication over addition or subtraction inside parentheses.
In this exercise, we are multiplying $a + 5$ by $a^2 - 2a + 4$.
To apply the distributive property, start by multiplying each term in the first polynomial by every term in the second polynomial. This will give you a set of partial products, which we later add together.

Combining Like Terms

Once we have our partial products, the next step is to combine like terms. Like terms have the same variable raised to the same power. For instance:
$a^2$ and $5a^2$ are like terms because they both have the same variable, $a$, raised to the second power.
To combine like terms, simply add or subtract the coefficients (numerical parts) of those terms.
In our problem, we have the terms $a^3, -2a^2, 4a, 5a^2, -10a,$ and $20$. Upon combining like terms:

$-2a^2$ and $5a^2$ yield $3a^2$
$4a$ and $-10a$ yield $-6a$

Thus, the expression simplifies to $a^3 + 3a^2 - 6a + 20$.

Polynomial Expansion

Polynomial expansion involves distributing each term of one polynomial to every term of another polynomial. As a result, we break down complex expressions into more manageable parts.
Here's a clearer walk-through of our example:
Start by distributing $a$ over $a^2 - 2a + 4$:

$a \times a^2 = a^3$
$a \times (-2a) = -2a^2$
$a \times 4 = 4a$

Then distribute $5$ over $a^2 - 2a + 4$:

$5 \times a^2 = 5a^2$
$5 \times (-2a) = -10a$
$5 \times 4 = 20$

This shows that expanding a polynomial expression step-by-step makes complex multiplication straightforward and manageable.

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91影视

Multiply the polynomials. $$(a+5)\left(a^{2}-2 a+4\right)$$

Short Answer

Step by step solution

Distribute the first term

Distribute the second term

Combine like terms

Key Concepts

Distributive Property

Combining Like Terms

Polynomial Expansion

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