Problem 19 Factor each trinomial completely... [FREE SOLUTION]

Chapter 6: Problem 19

Factor each trinomial completely. If a polynomial can't be factored, write "prime." See Examples I through 8 . $$ 10 t-24+t^{2} $$

Short Answer

Expert verified

The factored form is $(t + 12)(t - 2)$.

Step by step solution

Write the Polynomial in Standard Form

To begin factoring the trinomial, we first need to express it in standard quadratic form $ax^2 + bx + c$. Rearranging the given expression $t^2 + 10t - 24$ gives us the trinomial in standard form: $t^2 + 10t - 24$. This puts the coefficients in their respective places: $a = 1$, $b = 10$, and $c = -24$.

Identify a Pair of Numbers

Next, we need to find two numbers that multiply to $a \cdot c$ (i.e., $1 \cdot -24 = -24$) and add up to $b$ (i.e., 10). The numbers that satisfy both conditions are 12 and -2 because $12 \times (-2) = -24$ and $12 + (-2) = 10$.

Rewrite the Middle Term

Using the numbers found in the previous step (12 and -2), we split the middle term (10t) of the trinomial into two separate terms. Rewrite the trinomial as $t^2 + 12t - 2t - 24$.

Factor by Grouping

Now, we factor by grouping. Take the first two terms $t^2 + 12t$ and factor out $t$, giving us $t(t + 12)$. Then, take the last two terms $-2t - 24$ and factor out $-2$, giving us $-2(t + 12)$.

Extract the Common Factor

Notice that $(t + 12)$ is a common factor in both groups. Factor $(t + 12)$ out to get $(t + 12)(t - 2)$. This is the completely factored form of the original trinomial.

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Key Concepts

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

Quadratic Equation

A quadratic equation is a polynomial equation of degree 2. That means the highest exponent on the variable is 2. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, with $a eq 0$.

In the quadratic expression from the exercise, we start with $t^2 + 10t - 24$. This is already in a standard quadratic form, with $a = 1$, $b = 10$, and $c = -24$.

The $t^2$ term tells us the equation is quadratic.
The linear term, 10t, contributes to the equation's shape and direction.
The constant term $-24$ shifts the parabola up or down the y-axis.

Understanding the structure of the quadratic equation helps in choosing the right method to solve or factor it. In our case, factoring was the chosen method to simplify and solve it.

Polynomial Factoring

Polynomial factoring involves expressing a polynomial as a product of its factors. In simpler terms, it's about breaking down a polynomial into simpler parts that, when multiplied together, give back the original polynomial.

Factoring is very helpful in many algebraic processes, especially when solving quadratic equations. When factoring trinomials鈥攍ike $t^2 + 10t - 24$鈥攖he goal is to find two binomials whose product is the original trinomial.

In our context, after rewriting the polynomial in standard form, we found numbers that satisfy two conditions: multiplying to $a \cdot c$ and adding to $b$.
In this case, the numbers found were 12 and -2, which allowed a breakdown of the middle term for easier factoring.

Polynomials can be broken apart through various techniques; however, understanding the structure and relationships of coefficients and terms is key to choosing the right approach.

Factoring by Grouping

Factoring by grouping is a technique used to factor polynomials, typically when there are four terms. It involves arranging terms in such a way that certain groups of terms share a common factor.

In the given quadratic Polynomial, once the middle term was split into 12t and -2t, the polynomial became $t^2 + 12t - 2t - 24$.

This four-term polynomial can be split into two groups: $t^2 + 12t$ and $-2t - 24$.
Factoring out a common factor from each group gave $t(t + 12)$ and $-2(t + 12)$.
Recognizing $t+12$ as a common factor in both groups allows for further simplification to $(t + 12)(t - 2)$.

The trick with grouping is spotting how terms can be combined and factored. It streamlines the process, turning a complex polynomial into a simpler product of binomials.

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91影视

Factor each trinomial completely. If a polynomial can't be factored, write "prime." See Examples I through 8 . $$ 10 t-24+t^{2} $$

Short Answer

Step by step solution

Write the Polynomial in Standard Form

Identify a Pair of Numbers

Rewrite the Middle Term

Factor by Grouping

Extract the Common Factor

Key Concepts

Quadratic Equation

Polynomial Factoring

Factoring by Grouping

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