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Answer :
To solve the equation [tex]\(x^3 - 13x^2 + 47x - 35 = 0\)[/tex] given that 1 is a zero of [tex]\(f(x)\)[/tex], we can follow these steps:
1. Use the Given Zero: Since 1 is a zero of the polynomial, it means [tex]\(f(1) = 0\)[/tex]. This implies that [tex]\((x - 1)\)[/tex] is a factor of the polynomial.
2. Factor the Polynomial: We can perform polynomial division to divide the given polynomial by [tex]\((x - 1)\)[/tex] to find the other factors.
3. Divide the Polynomial: Dividing [tex]\(x^3 - 13x^2 + 47x - 35\)[/tex] by [tex]\((x - 1)\)[/tex] will simplify the polynomial. After performing the division, the quotient is [tex]\(x^2 - 12x + 35\)[/tex].
4. Factor the Quadratic: Next, we need to factor the quadratic [tex]\(x^2 - 12x + 35\)[/tex]. We look for two numbers that multiply to 35 and add to -12. These numbers are -7 and -5.
5. Complete the Factorization: Thus, the factorization of [tex]\(x^2 - 12x + 35\)[/tex] is [tex]\((x - 7)(x - 5)\)[/tex].
6. Combine the Factors: Now, combining all the factors, we have:
[tex]\[
(x - 1)(x - 7)(x - 5) = 0
\][/tex]
7. Solve for x: To find the solutions, set each factor equal to zero:
- [tex]\(x - 1 = 0 \rightarrow x = 1\)[/tex]
- [tex]\(x - 7 = 0 \rightarrow x = 7\)[/tex]
- [tex]\(x - 5 = 0 \rightarrow x = 5\)[/tex]
Therefore, the solutions to the equation [tex]\(x^3 - 13x^2 + 47x - 35 = 0\)[/tex] are [tex]\(x = 1\)[/tex], [tex]\(x = 7\)[/tex], and [tex]\(x = 5\)[/tex].
1. Use the Given Zero: Since 1 is a zero of the polynomial, it means [tex]\(f(1) = 0\)[/tex]. This implies that [tex]\((x - 1)\)[/tex] is a factor of the polynomial.
2. Factor the Polynomial: We can perform polynomial division to divide the given polynomial by [tex]\((x - 1)\)[/tex] to find the other factors.
3. Divide the Polynomial: Dividing [tex]\(x^3 - 13x^2 + 47x - 35\)[/tex] by [tex]\((x - 1)\)[/tex] will simplify the polynomial. After performing the division, the quotient is [tex]\(x^2 - 12x + 35\)[/tex].
4. Factor the Quadratic: Next, we need to factor the quadratic [tex]\(x^2 - 12x + 35\)[/tex]. We look for two numbers that multiply to 35 and add to -12. These numbers are -7 and -5.
5. Complete the Factorization: Thus, the factorization of [tex]\(x^2 - 12x + 35\)[/tex] is [tex]\((x - 7)(x - 5)\)[/tex].
6. Combine the Factors: Now, combining all the factors, we have:
[tex]\[
(x - 1)(x - 7)(x - 5) = 0
\][/tex]
7. Solve for x: To find the solutions, set each factor equal to zero:
- [tex]\(x - 1 = 0 \rightarrow x = 1\)[/tex]
- [tex]\(x - 7 = 0 \rightarrow x = 7\)[/tex]
- [tex]\(x - 5 = 0 \rightarrow x = 5\)[/tex]
Therefore, the solutions to the equation [tex]\(x^3 - 13x^2 + 47x - 35 = 0\)[/tex] are [tex]\(x = 1\)[/tex], [tex]\(x = 7\)[/tex], and [tex]\(x = 5\)[/tex].
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