Join IDNLearn.com today and start getting the answers you've been searching for. Ask your questions and receive reliable, detailed answers from our dedicated community of experts.
Sagot :
To find the correct factor pairs of the polynomial given in the problem, we can substitute the polynomial with its likely factors based on the given options. We want to ensure that the polynomial [tex]\( x^2 - x - 6 \)[/tex] is correctly represented.
Given polynomial:
[tex]\[ x^2 - x - 6 \][/tex]
Let's rewrite our options and multiply them to check which factors produce the polynomial:
1. Factors: (x - 1) and (x + 3)
[tex]\[ (x - 1)(x + 3) = x(x + 3) - 1(x + 3) = x^2 + 3x - x - 3 = x^2 + 2x - 3 \][/tex]
2. Factors: (x + 1) and (x - 3)
[tex]\[ (x + 1)(x - 3) = x(x - 3) + 1(x - 3) = x^2 - 3x + x - 3 = x^2 - 2x - 3 \][/tex]
3. Factors: (x - 2) and (x + 3)
[tex]\[ (x - 2)(x + 3) = x(x + 3) - 2(x + 3) = x^2 + 3x - 2x - 6 = x^2 + x - 6 \][/tex]
4. Factors: (x + 2) and (x - 3)
[tex]\[ (x + 2)(x - 3) = x(x - 3) + 2(x - 3) = x^2 - 3x + 2x - 6 = x^2 - x - 6 \][/tex]
Now, let's see which one of these expressions matches the given polynomial [tex]\( x^2 - x - 6 \)[/tex]:
- The first pair, [tex]\((x - 1)(x + 3)\)[/tex], results in [tex]\( x^2 + 2x - 3 \)[/tex]
- The second pair, [tex]\((x + 1)(x - 3)\)[/tex], results in [tex]\( x^2 - 2x - 3 \)[/tex]
- The third pair, [tex]\((x - 2)(x + 3)\)[/tex], results in [tex]\( x^2 + x - 6 \)[/tex]
- The fourth pair, [tex]\((x + 2)(x - 3)\)[/tex], results in [tex]\( x^2 - x - 6 \)[/tex]
Thus, the correct pair of factors for the polynomial [tex]\( x^2 - x - 6 \)[/tex] is:
[tex]\[ (x + 2) \text{ and } (x - 3) \][/tex]
Given polynomial:
[tex]\[ x^2 - x - 6 \][/tex]
Let's rewrite our options and multiply them to check which factors produce the polynomial:
1. Factors: (x - 1) and (x + 3)
[tex]\[ (x - 1)(x + 3) = x(x + 3) - 1(x + 3) = x^2 + 3x - x - 3 = x^2 + 2x - 3 \][/tex]
2. Factors: (x + 1) and (x - 3)
[tex]\[ (x + 1)(x - 3) = x(x - 3) + 1(x - 3) = x^2 - 3x + x - 3 = x^2 - 2x - 3 \][/tex]
3. Factors: (x - 2) and (x + 3)
[tex]\[ (x - 2)(x + 3) = x(x + 3) - 2(x + 3) = x^2 + 3x - 2x - 6 = x^2 + x - 6 \][/tex]
4. Factors: (x + 2) and (x - 3)
[tex]\[ (x + 2)(x - 3) = x(x - 3) + 2(x - 3) = x^2 - 3x + 2x - 6 = x^2 - x - 6 \][/tex]
Now, let's see which one of these expressions matches the given polynomial [tex]\( x^2 - x - 6 \)[/tex]:
- The first pair, [tex]\((x - 1)(x + 3)\)[/tex], results in [tex]\( x^2 + 2x - 3 \)[/tex]
- The second pair, [tex]\((x + 1)(x - 3)\)[/tex], results in [tex]\( x^2 - 2x - 3 \)[/tex]
- The third pair, [tex]\((x - 2)(x + 3)\)[/tex], results in [tex]\( x^2 + x - 6 \)[/tex]
- The fourth pair, [tex]\((x + 2)(x - 3)\)[/tex], results in [tex]\( x^2 - x - 6 \)[/tex]
Thus, the correct pair of factors for the polynomial [tex]\( x^2 - x - 6 \)[/tex] is:
[tex]\[ (x + 2) \text{ and } (x - 3) \][/tex]
Your engagement is important to us. Keep sharing your knowledge and experiences. Let's create a learning environment that is both enjoyable and beneficial. For dependable answers, trust IDNLearn.com. Thank you for visiting, and we look forward to assisting you again.