Problem 35
Question
For the following problems, factor the trinomials when possible. $$ 4 x^{2}+12 x+8 $$
Step-by-Step Solution
Verified Answer
Answer: The factored form of the trinomial $$4x^2 + 12x + 8$$ is $$4(x + 1)(x + 2)$$.
1Step 1: Identify the trinomial and find the Greatest Common Factor (GCF)
The given trinomial is $$4x^2 + 12x + 8$$. Let's find the GCF of the coefficients (4, 12, and 8). The GCF of 4, 12, and 8 is 4.
2Step 2: Factor out the GCF
We found that the GCF is 4, so now we will factor out 4 from the given trinomial.
$$
4x^2 + 12x + 8 = 4(x^2 + 3x + 2)
$$
Now, we have a simpler trinomial $$x^2 + 3x + 2$$ to factor further.
3Step 3: Factor the trinomial within the parentheses
To factor $$x^2 + 3x + 2$$, we need to find two binomials whose product is equal to the given trinomial. The binomials should have the form $$(x + p)(x + q)$$ where p and q are numbers such that $$p \times q = 2$$ and $$p + q = 3$$. By trial and error, we find that $$p = 1$$ and $$q = 2$$ satisfy these conditions.
4Step 4: Write the factored form of the trinomial
After finding the values of p and q, we can now write the factored form of the trinomial inside the parentheses:
$$
x^2 + 3x + 2 = (x + 1)(x + 2)
$$
So, the fully factored form of the given trinomial is:
$$
4x^2 + 12x + 8 = 4(x + 1)(x + 2)
$$
Key Concepts
Greatest Common FactorBinomialsTrinomialPolynomials
Greatest Common Factor
When working with trinomials, finding the greatest common factor (GCF) is a key step to simplifying the expression. The GCF is the largest factor that divides each term of the polynomial without leaving any remainder. To identify it, follow these simple steps:
Factoring out the GCF essentially involves dividing each term of the polynomial by this common factor. This simplifies the polynomial to make further factoring easier, transforming it into a more manageable form.
- List out the factors of each coefficient in the polynomial.
- Identify the largest factor that appears in all lists.
Factoring out the GCF essentially involves dividing each term of the polynomial by this common factor. This simplifies the polynomial to make further factoring easier, transforming it into a more manageable form.
Binomials
Binomials are algebraic expressions containing exactly two terms, connected by either a positive or a negative sign. Understanding binomials is essential because many polynomials, including trinomials, can be expressed as products of binomials.
The process of factoring a trinomial often involves writing it as a product of two binomials. For example, in the expression \(x^2 + 3x + 2\), we are looking for two numbers that multiply to the constant term (2) and add to the middle coefficient (3). These numbers, \(p\) and \(q\), help form the binomials \((x + p)(x + q)\).
The process of factoring a trinomial often involves writing it as a product of two binomials. For example, in the expression \(x^2 + 3x + 2\), we are looking for two numbers that multiply to the constant term (2) and add to the middle coefficient (3). These numbers, \(p\) and \(q\), help form the binomials \((x + p)(x + q)\).
- Example: When \(p = 1\) and \(q = 2\), the binomial factors are \((x + 1)(x + 2)\).
Trinomial
A trinomial is a type of polynomial that contains three terms. The typical format is \(ax^2 + bx + c\). Factoring trinomials is a common algebraic task, especially when simplifying expressions or solving equations.
- Quadratic Trinomials: These are trinomials of the form \(x^2 + bx + c\) and are often factored into products of two binomials.
- Steps to Factor: Identify two numbers that multiply to \(c\) and add to \(b\). Use these to split the middle term and group the expression for easier factoring.
Polynomials
Polynomials are algebraic expressions made up of terms consisting of variables and coefficients. They can have various numbers of terms and can be simple or complex, such as binomials and trinomials.
Understanding how to manipulate polynomials is crucial in algebra. They serve as the foundation for many tasks, such as simplifying expressions and solving equations. Polynomials can also be divided into categories such as:
Understanding how to manipulate polynomials is crucial in algebra. They serve as the foundation for many tasks, such as simplifying expressions and solving equations. Polynomials can also be divided into categories such as:
- Monomials: Polynomials with one term, like 3x or 5.
- Binomials: Polynomials with two terms.
- Trinomials: Polynomials with three terms.
- Higher-degree Polynomials: Polynomials with more than three terms or a degree greater than 2.
Other exercises in this chapter
Problem 34
For the following problems, factor the polynomials. $$ 30 a^{2} b^{2}+40 a^{2} b^{2}+50 a^{2} b^{2} $$
View solution Problem 34
In the following problems, the first quantity represents the product and the second quantity represents a factor of that product. Find the other factor. $$ (a+3
View solution Problem 35
For the following problems, use the grouping method to factor the polynomials. Some may not be factorable. $$ a^{2}-7 a+a b-7 b $$
View solution Problem 35
For the following problems, factor the binomials. $$ x^{4} y^{4}-9 a^{2} $$
View solution