Davide wants to find the zeros of the polynomial p(x)=x3−2x2−9x+18. His work is shown below.
Step 1: p(x)=(x+3)(x−3)(x−2)
Step 2: x+3=0
x−3=0
x−2=0
Step 3: x=2, x=3, x=−3
Explain why Davide’s work shows that 2, 3, and −3 are zeros of p(x) by matching each step with the appropriate justification.
Step 1
Step 2
Step 3
The choices to fill each step are
A)Davide gets rid of the x in each of the equations from Step 2 by setting x=0 . The zeros of a polynomial occur when x=0 .
B)Davide sets each linear factor equal to 0 because the x -value for the zero of a polynomial is 0 .
C)Davide solves for x in each of the equations from Step 2. The zeros of a polynomial are the x -values for which p(x)=0 .
D)Davide factors the polynomial p(x) into linear factors using factoring by grouping and the difference of squares.
E)Davide factors the polynomial p(x) into linear factors using the difference of cubes method.
F)Davide sets each linear factor equal to 0 because of the Zero Product Property.
I said
Step 1: Davide gets rid of the x in each of the equations from Step 2 by setting x=0. The zeros of a polynomial occur when x=0.
Step 2:Davide sets each linear factor equal to 0 because of the Zero Product Property.
Step 3:Davide factors the polynomial p(x) into linear factors using factoring by grouping and the difference of squares.
Please help me and give me feedback. Thank you
step1 is false. The zeroes of p(x) occur when p=0
I like D,F,C
I can verify that the answer is indeed:
Step 1:D
Step 2:F
Step 3:C
Your answers for Step 1 and Step 2 are incorrect, but your answer for Step 3 is correct. Let's go through each step:
Step 1: Davide factors the polynomial p(x) into linear factors using factoring techniques such as factoring by grouping and the difference of squares. In this case, the factored form of p(x) is (x+3)(x-3)(x-2). This step is necessary to get the polynomial in the form of linear factors.
Step 2: Davide sets each linear factor equal to 0 because of the Zero Product Property. The Zero Product Property states that if a product of factors equals zero, then at least one of the factors must be zero. In this case, since p(x) is the product (x+3)(x-3)(x-2), Davide sets each factor equal to zero separately: x+3=0, x-3=0, and x-2=0.
Step 3: Davide solves for x in each of the equations from Step 2. By solving these equations, he finds the values of x that make each factor equal to zero. The solutions for each equation are x=-3, x=3, and x=2. Since these values make each factor zero, they are called the zeros of the polynomial.
So, the correct answers are:
Step 1: Davide factors the polynomial p(x) into linear factors using factoring techniques such as factoring by grouping and the difference of squares.
Step 2: Davide sets each linear factor equal to zero because of the Zero Product Property.
Step 3: Davide solves for x in each of the equations from Step 2. The zeros of a polynomial are the x-values for which p(x) is equal to zero.
Your explanations for steps 1 and 2 are mostly correct, but there are some minor inaccuracies. Here are the correct explanations for each step:
Step 1: Davide factors the polynomial p(x) into linear factors using factoring. This step is shown as p(x) = (x + 3)(x - 3)(x - 2). Factoring allows us to express the polynomial as a product of linear factors.
Step 2: Davide sets each linear factor equal to 0 because of the Zero Product Property. This property states that if a product of factors is equal to zero, then at least one of the factors must be zero. In this case, setting each linear factor equal to zero will give us the values of x that make the polynomial equal to zero.
Step 3: Davide solves for x in each of the equations from Step 2. Setting each linear factor equal to zero allows us to solve for x. Solving each equation gives us the values of x that satisfy the equation and make the polynomial equal to zero. In this case, x = 2, x = 3, and x = -3 are the solutions.
So the correct answers for each step are:
Step 1: D) Davide factors the polynomial p(x) into linear factors using factoring.
Step 2: F) Davide sets each linear factor equal to 0 because of the Zero Product Property.
Step 3: C) Davide solves for x in each of the equations from Step 2.
Your explanations are mostly accurate, but the feedback is to provide more precise and accurate explanations for each step.