Problem 36 $f(x)=3(x-10)(x+4)$... [FREE SOLUTION]

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BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Ron Larson, Laurie Boswell

$Math Studyset 91影视 Explanations$ Math

1 Edition

Chapter 4: Problem 36

$f(x)=3(x-10)(x+4)$

Short Answer

Expert verified

The simplified expression for the function is $f(x) = 3x^2 - 18x - 120$.

Step by step solution

Distribute the Inner and Outer Terms

First distribute the $x$ in the expression $x-10$ times the two terms in the parentheses $x+4$. This gives $(x)(x) + (x)(4)$ which simplifies to $x^2 + 4x$.

Distribute the Last Terms

Next, distribute the $-10$ in the expression $x-10$ times the two terms in the parentheses $x+4$. This gives $(-10)(x) + (-10)(4)$ which simplifies to $-10x - 40$.

Combine Like Terms and Distribute the 3

Combine like terms $x^2 + 4x - 10x - 40$ going to $x^2 - 6x - 40$. Then distribute the 3 across all terms in the expression $3(x^2 - 6x - 40)$ to get the final expression $3x^2 - 18x - 120$.

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Key Concepts

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

Factoring

Factoring is a fundamental process in algebra used to simplify polynomial expressions. It involves rewiring an expression as a product of its factors. Think of it as breaking down a number or expression into its building blocks. For instance, in the expression $3(x-10)(x+4)$, the polynomial

is factored into three separate components: $3$, $x-10$, and $x+4$.
These components can be further expanded into a polynomial by multiplying.

Factoring makes it easier to simplify, solve, or graph the polynomial by identifying its zeroes directly from the factors. The zeroes of the polynomial are derived from setting each factor equal to zero.

Distribution

Distribution, or the distributive property, is a central concept in algebra that allows us to multiply numbers or expressions in parentheses by a number or another expression outside the parentheses. To distribute means to spread the factor outside the parentheses over each term inside the bracket. In the case of the expression $3(x-10)(x+4)$:

We first distribute each term in $(x+4)$ to the terms in $(x-10)$ which involves multiplying $x\cdot x$, $x\cdot 4$, $-10\cdot x$, and $-10\cdot 4$.
This results in the expression $x^2 + 4x - 10x - 40$.

Finally, the number$3$ outside the parentheses is distributed across the new polynomial to get $3x^2 - 18x - 120$. Think of it as sharing or spreading multiplication across additions and subtractions within parentheses.

Combining Like Terms

Combining like terms is a crucial technique to simplify polynomial expressions. Like terms have the same variable raised to the same power. When simplifying, you add or subtract their coefficients:

In $x^2 + 4x - 10x - 40$, the terms $4x$ and $-10x$ are like terms.
By combining them, we perform the operation $4x - 10x$, which results in $-6x$.

After this step, the expression becomes $x^2 - 6x - 40$. Combining like terms is key in polynomial simplification as it reduces the number of terms and simplifies the expression.

Quadratic Expressions

Quadratic expressions are polynomials where the highest exponent of the variable is 2. They typically appear in the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. In our exercise, the quadratic expression resulted from distributing and combining terms from $(x-10)(x+4)$, yielding $x^2 - 6x - 40$. After distributing the 3, we get $3x^2 - 18x - 120$:

This expression, $3x^2 - 18x - 120$, is quadratic because the term with the highest degree is $x^2$.
Quadratic expressions can be graphed as parabolas and are fundamental in solving quadratic equations, which determine where the graph crosses the x-axis (roots or solutions).

Understanding quadratic expressions is essential as they commonly appear in various real-world and mathematical problems.

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91影视

\(f(x)=3(x-10)(x+4)\)

Short Answer

Step by step solution

Distribute the Inner and Outer Terms

Distribute the Last Terms

Combine Like Terms and Distribute the 3

Key Concepts

Factoring

Distribution

Combining Like Terms

Quadratic Expressions

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