Get clear, concise, and accurate answers to your questions on IDNLearn.com. Explore thousands of verified answers from experts and find the solutions you need, no matter the topic.
Sagot :
To factor the polynomial [tex]\( x^3 + 1 \)[/tex] with integer coefficients, follow these steps:
1. Recognize the form of the polynomial:
The polynomial [tex]\( x^3 + 1 \)[/tex] is a sum of cubes. A sum of cubes can be factored using the formula:
[tex]\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \][/tex]
2. Identify [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
In our polynomial [tex]\( x^3 + 1 \)[/tex], we can identify [tex]\( a = x \)[/tex] and [tex]\( b = 1 \)[/tex].
3. Apply the sum of cubes formula:
Substitute [tex]\( a = x \)[/tex] and [tex]\( b = 1 \)[/tex] into the formula:
[tex]\[ x^3 + 1^3 = (x + 1)(x^2 - x \cdot 1 + 1^2) \][/tex]
4. Simplify the expression:
Perform the multiplications and additions inside the second factor:
[tex]\[ x^2 - x \cdot 1 + 1^2 = x^2 - x + 1 \][/tex]
5. Combine the factors:
Substitute the simplified second factor back:
[tex]\[ x^3 + 1 = (x + 1)(x^2 - x + 1) \][/tex]
Thus, the factorization of [tex]\( x^3 + 1 \)[/tex] with integer coefficients is:
[tex]\[ x^3 + 1 = (x + 1)(x^2 - x + 1) \][/tex]
This is the fully factored form of the polynomial with integer coefficients.
1. Recognize the form of the polynomial:
The polynomial [tex]\( x^3 + 1 \)[/tex] is a sum of cubes. A sum of cubes can be factored using the formula:
[tex]\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \][/tex]
2. Identify [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
In our polynomial [tex]\( x^3 + 1 \)[/tex], we can identify [tex]\( a = x \)[/tex] and [tex]\( b = 1 \)[/tex].
3. Apply the sum of cubes formula:
Substitute [tex]\( a = x \)[/tex] and [tex]\( b = 1 \)[/tex] into the formula:
[tex]\[ x^3 + 1^3 = (x + 1)(x^2 - x \cdot 1 + 1^2) \][/tex]
4. Simplify the expression:
Perform the multiplications and additions inside the second factor:
[tex]\[ x^2 - x \cdot 1 + 1^2 = x^2 - x + 1 \][/tex]
5. Combine the factors:
Substitute the simplified second factor back:
[tex]\[ x^3 + 1 = (x + 1)(x^2 - x + 1) \][/tex]
Thus, the factorization of [tex]\( x^3 + 1 \)[/tex] with integer coefficients is:
[tex]\[ x^3 + 1 = (x + 1)(x^2 - x + 1) \][/tex]
This is the fully factored form of the polynomial with integer coefficients.
We appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. For precise answers, trust IDNLearn.com. Thank you for visiting, and we look forward to helping you again soon.