Problem 51
Question
CHECK FACTORING. Use a graphing calculator to determine if each polynomial is factored correctly. Write yes or no. If the polynomial is not factored correctly, find the correct factorization. $$ 3 x^{2}+5 x+2 \stackrel{?}{=}(3 x+2)(x+1) $$
Step-by-Step Solution
Verified Answer
Yes, the polynomial is factored correctly as \((3x+2)(x+1)\).
1Step 1: Expand the Given Factors
Let's expand the expression \((3x + 2)(x + 1)\) to check if it equals the original polynomial. We will use the distributive property (FOIL method) to expand it. \[(3x + 2)(x + 1) = 3x(x) + 3x(1) + 2(x) + 2(1) = 3x^2 + 3x + 2x + 2 = 3x^2 + 5x + 2\].
2Step 2: Compare with the Original Polynomial
Now, compare the expanded polynomial \(3x^2 + 5x + 2\) from Step 1 with the original polynomial \(3x^2 + 5x + 2\). Since they are equal, \((3x + 2)(x + 1)\) is indeed the correct factorization of the polynomial.
3Step 3: Verify with a Graphing Calculator
To ensure everything is correct, plot both the original polynomial \(3x^2 + 5x + 2\) and the factored polynomial \((3x + 2)(x + 1)\) on a graphing calculator. Both should produce the same graph, confirming that the factorization is correct.
Key Concepts
Polynomial ExpansionDistributive PropertyGraphing CalculatorFOIL Method
Polynomial Expansion
Polynomial expansion is the process of converting a factored expression into a simpler form by multiplying it out. It’s like unfolding a compact idea into its full expression. This process is crucial for verifying the correctness of a factorization.
When you’re given a factored polynomial, expanding it helps you see whether you end up with your original expression. In the exercise provided, we expanded \((3x + 2)(x + 1)\), using multiplication to determine whether it equated to \(3x^2 + 5x + 2\).
Expanding helps test accuracy. If the expanded form equals the original expression, the factorization is correct. This practice is not only essential in algebra but serves as a foundation for solving many types of mathematical problems.
When you’re given a factored polynomial, expanding it helps you see whether you end up with your original expression. In the exercise provided, we expanded \((3x + 2)(x + 1)\), using multiplication to determine whether it equated to \(3x^2 + 5x + 2\).
Expanding helps test accuracy. If the expanded form equals the original expression, the factorization is correct. This practice is not only essential in algebra but serves as a foundation for solving many types of mathematical problems.
Distributive Property
The distributive property is a fundamental mathematical principle used when expanding polynomials. It states that for any numbers \(a\), \(b\), and \(c\), the equation \(a(b + c) = ab + ac\) holds true.
This property helps break down complex multiplications into more manageable parts. So, in our problem, we applied the distributive property during polynomial expansion by multiplying each term in one polynomial by every term in the other.
For example, in \((3x + 2)(x + 1)\), we distribute \(3x\) across \((x + 1)\) and separately distribute \(2\) across \((x + 1)\).
So we have:
This property helps break down complex multiplications into more manageable parts. So, in our problem, we applied the distributive property during polynomial expansion by multiplying each term in one polynomial by every term in the other.
For example, in \((3x + 2)(x + 1)\), we distribute \(3x\) across \((x + 1)\) and separately distribute \(2\) across \((x + 1)\).
So we have:
- \(3x \times x = 3x^2\)
- \(3x \times 1 = 3x\)
- \(2 \times x = 2x\)
- \(2 \times 1 = 2\)
Graphing Calculator
A graphing calculator is a versatile tool that can visually confirm the correctness of a polynomial's factorization. By graphing both the original polynomial and its supposed factorization, you can see if they produce the same graph.
This helps verify that the expanded expression matches the originally factored polynomial in practice. When both expressions are graphed and the plots overlap, it indicates that the factored form is correct.
Using a graphing calculator involves:
This helps verify that the expanded expression matches the originally factored polynomial in practice. When both expressions are graphed and the plots overlap, it indicates that the factored form is correct.
Using a graphing calculator involves:
- Inputting the original polynomial (\(3x^2 + 5x + 2\)).
- Inputting the factored polynomial \((3x + 2)(x + 1)\).
- Checking if both graphs coincide.
FOIL Method
The FOIL method is a specialized case of the distributive property used when multiplying two binomials. FOIL stands for First, Outer, Inner, Last, representing the order in which you multiply the terms of the binomials.
In the exercise, the expression \((3x + 2)(x + 1)\) involves the following steps:
The beauty of the FOIL method is its structured reliability, making polynomial multiplication simpler and systematic.
In the exercise, the expression \((3x + 2)(x + 1)\) involves the following steps:
- First: Multiply the first terms in each binomial: \(3x \times x = 3x^2\).
- Outer: Multiply the outer terms: \(3x \times 1 = 3x\).
- Inner: Multiply the inner terms: \(2 \times x = 2x\).
- Last: Multiply the last terms: \(2 \times 1 = 2\).
The beauty of the FOIL method is its structured reliability, making polynomial multiplication simpler and systematic.
Other exercises in this chapter
Problem 51
If \(p(x)=2 x^{2}-5 x+4\) and \(r(x)=3 x^{3}-x^{2}-2,\) find each value. $$ 2\left[p\left(x^{2}+1\right)\right]-3 r(x-1) $$
View solution Problem 51
How many negative real zeros does \(f(x)=x^{5}-2 x^{4}-4 x^{3}+\) \(4 x^{2}-5 x+6\) have? A. 3 B. 2 C. 1 D. 0
View solution Problem 51
Simplify. $$ (1+4 c)^{2} $$
View solution Problem 51
Simplify. $$ (a-b)^{2} $$
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