Find the best solutions to your problems with the help of IDNLearn.com's experts. Discover reliable answers to your questions with our extensive database of expert knowledge.
Sagot :
To construct a polynomial in standard form given its roots, we need to utilize the fact that if [tex]\( r \)[/tex] is a root of the polynomial, then [tex]\( (x - r) \)[/tex] is a factor of the polynomial. The polynomial is the product of these factors.
Given the roots [tex]\( x = 5 \)[/tex] with multiplicity 1 and [tex]\( x = -4 \)[/tex] with multiplicity 2, we construct the polynomial as follows:
### Step-by-Step Solution
1. Identify the Factors:
- For the root [tex]\( x = 5 \)[/tex]: the factor is [tex]\( (x - 5) \)[/tex]
- For the root [tex]\( x = -4 \)[/tex] with multiplicity 2: the factor is [tex]\( (x + 4) \)[/tex] but raised to the power of 2 to account for the multiplicity. Thus, the factor is [tex]\( (x + 4)^2 \)[/tex].
2. Write the Polynomial as a Product of Factors:
[tex]\[ P(x) = (x - 5)(x + 4)^2 \][/tex]
3. Expand the Polynomial:
First, expand [tex]\( (x + 4)^2 \)[/tex]:
[tex]\[ (x + 4)^2 = (x + 4)(x + 4) = x^2 + 8x + 16 \][/tex]
Now, multiply this result by [tex]\( (x - 5) \)[/tex]:
[tex]\[ P(x) = (x - 5)(x^2 + 8x + 16) \][/tex]
4. Distribute [tex]\( (x - 5) \)[/tex] across [tex]\( (x^2 + 8x + 16) \)[/tex]:
[tex]\[ P(x) = x(x^2 + 8x + 16) - 5(x^2 + 8x + 16) \][/tex]
Let's distribute [tex]\( x \)[/tex]:
[tex]\[ x(x^2 + 8x + 16) = x^3 + 8x^2 + 16x \][/tex]
Now, distribute [tex]\( -5 \)[/tex]:
[tex]\[ -5(x^2 + 8x + 16) = -5x^2 - 40x - 80 \][/tex]
Combine the two results together:
[tex]\[ P(x) = x^3 + 8x^2 + 16x - 5x^2 - 40x - 80 \][/tex]
5. Combine Like Terms:
[tex]\[ P(x) = x^3 + 3x^2 - 24x - 80 \][/tex]
### The Polynomial in Standard Form:
[tex]\[ P(x) = x^3 + 3x^2 - 24x - 80 \][/tex]
Thus, the polynomial that fits the given roots and their multiplicities is:
- [tex]\( x = 5 \)[/tex] (multiplicity 1)
- [tex]\( x = -4 \)[/tex] (multiplicity 2)
is
[tex]\[ \boxed{x^3 + 3x^2 - 24x - 80} \][/tex]
Given the roots [tex]\( x = 5 \)[/tex] with multiplicity 1 and [tex]\( x = -4 \)[/tex] with multiplicity 2, we construct the polynomial as follows:
### Step-by-Step Solution
1. Identify the Factors:
- For the root [tex]\( x = 5 \)[/tex]: the factor is [tex]\( (x - 5) \)[/tex]
- For the root [tex]\( x = -4 \)[/tex] with multiplicity 2: the factor is [tex]\( (x + 4) \)[/tex] but raised to the power of 2 to account for the multiplicity. Thus, the factor is [tex]\( (x + 4)^2 \)[/tex].
2. Write the Polynomial as a Product of Factors:
[tex]\[ P(x) = (x - 5)(x + 4)^2 \][/tex]
3. Expand the Polynomial:
First, expand [tex]\( (x + 4)^2 \)[/tex]:
[tex]\[ (x + 4)^2 = (x + 4)(x + 4) = x^2 + 8x + 16 \][/tex]
Now, multiply this result by [tex]\( (x - 5) \)[/tex]:
[tex]\[ P(x) = (x - 5)(x^2 + 8x + 16) \][/tex]
4. Distribute [tex]\( (x - 5) \)[/tex] across [tex]\( (x^2 + 8x + 16) \)[/tex]:
[tex]\[ P(x) = x(x^2 + 8x + 16) - 5(x^2 + 8x + 16) \][/tex]
Let's distribute [tex]\( x \)[/tex]:
[tex]\[ x(x^2 + 8x + 16) = x^3 + 8x^2 + 16x \][/tex]
Now, distribute [tex]\( -5 \)[/tex]:
[tex]\[ -5(x^2 + 8x + 16) = -5x^2 - 40x - 80 \][/tex]
Combine the two results together:
[tex]\[ P(x) = x^3 + 8x^2 + 16x - 5x^2 - 40x - 80 \][/tex]
5. Combine Like Terms:
[tex]\[ P(x) = x^3 + 3x^2 - 24x - 80 \][/tex]
### The Polynomial in Standard Form:
[tex]\[ P(x) = x^3 + 3x^2 - 24x - 80 \][/tex]
Thus, the polynomial that fits the given roots and their multiplicities is:
- [tex]\( x = 5 \)[/tex] (multiplicity 1)
- [tex]\( x = -4 \)[/tex] (multiplicity 2)
is
[tex]\[ \boxed{x^3 + 3x^2 - 24x - 80} \][/tex]
Your participation means a lot to us. Keep sharing information and solutions. This community grows thanks to the amazing contributions from members like you. Thank you for choosing IDNLearn.com. We’re here to provide reliable answers, so please visit us again for more solutions.