Problem 78 Multiply. See Examples I through... [FREE SOLUTION]

Chapter 5: Problem 78

Multiply. See Examples I through II. $$ \left(13 x^{2}+1\right)(3 x+1) $$

Short Answer

Expert verified

The product is $39x^3 + 13x^2 + 3x + 1$.

Step by step solution

Apply the distributive property

Multiply each term in the first polynomial, $13x^2 + 1$, by each term in the second polynomial, $3x + 1$. This results in four products: $13x^2 \cdot 3x$, $13x^2 \cdot 1$, $1 \cdot 3x$, and $1 \cdot 1$.

Calculate the products

Perform each of the multiplications calculated in Step 1: 1. $13x^2 \cdot 3x = 39x^3$, 2. $13x^2 \cdot 1 = 13x^2$, 3. $1 \cdot 3x = 3x$, 4. $1 \cdot 1 = 1$.

Combine like terms

Write all products together and combine any like terms. The expression becomes: $39x^3 + 13x^2 + 3x + 1$. Since none of the terms can be combined further, this is the final simplified polynomial.

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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 a fundamental mathematical principle used frequently when multiplying polynomials. At its core, it means distributing, or spreading, each term of one expression across every term of another expression. When dealing with two polynomials, you multiply each term from the first polynomial with each term from the second.

For our exercise, you have $ (13x^2 + 1)(3x + 1) $. You apply the distributive property by multiplying every term of $13x^2 + 1$ with every term of $3x + 1$.

First, $13x^2$ is multiplied by $3x$ and $1$.
Next, $1$ is also multiplied by $3x$ and $1$.

This way, you ensure that all combinations of terms have been considered, generating the terms: $39x^3$, $13x^2$, $3x$, and $1$.

Combining Like Terms

Once all terms have been calculated using the distributive property, the next step is to combine like terms. Like terms are terms in an expression that have the same variable raised to the same power. These can be added or subtracted together to simplify the expression.

In our example, the products from the distributive step are $39x^3$, $13x^2$, $3x$, and $1$. Each of these terms has a different power of $x$: $x^3$, $x^2$, $x$, and a constant.

This means there are no terms with the same power of $x$ that can be combined.

So, in this particular case, the expression remains as $39x^3 + 13x^2 + 3x + 1$ because each term has a unique combination of coefficients and variable powers.

Simplifying Polynomials

Simplifying polynomials involves reducing the expression to its most compact form without changing its value. This process includes the steps we've talked about: using the distributive property and combining like terms. After multiplying and combining, you should always check if further simplification is possible.

One goal in simplifying a polynomial is to ensure it is free of unnecessary terms and in a form where no further simplification can occur.

In our problem, after performing the necessary operations, the polynomial $39x^3 + 13x^2 + 3x + 1$ has no like terms to combine, and it's already tidied up.
This means we've reached the simplest form of the polynomial expression originally given: $(13x^2 + 1)(3x + 1)$.

Having a neat and concise polynomial expression makes it easier to analyze and ensures you have understood all the transformations your polynomial has undergone.

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91影视

Multiply. See Examples I through II. $$ \left(13 x^{2}+1\right)(3 x+1) $$

Short Answer

Step by step solution

Apply the distributive property

Calculate the products

Combine like terms

Key Concepts

Distributive Property

Combining Like Terms

Simplifying Polynomials

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