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Answer :
To express the given polynomial [tex]\(625x^4 - 7,500x^3 + 33,750x^2 - 67,500x + 50,625\)[/tex] as [tex]\((ax + b)^4\)[/tex], we need to find suitable values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex].
1. Identify the Coefficient of the Highest Degree Term:
The highest degree term is [tex]\(625x^4\)[/tex]. Since the polynomial is in the form [tex]\((ax+b)^4\)[/tex], the coefficient of [tex]\(x^4\)[/tex] should be [tex]\(a^4\)[/tex]. Therefore:
[tex]\[
a^4 = 625
\][/tex]
To find [tex]\(a\)[/tex], we need to take the fourth root of 625:
[tex]\[
\sqrt[4]{625} = 5
\][/tex]
Thus, [tex]\(a = 5\)[/tex].
2. Determine the Constant Term:
The constant term of the expansion [tex]\((ax+b)^4\)[/tex] is [tex]\(b^4\)[/tex]. In the polynomial, this constant term is 50,625:
[tex]\[
b^4 = 50,625
\][/tex]
To find [tex]\(b\)[/tex], take the fourth root of 50,625:
[tex]\[
b = \sqrt[4]{50,625} = 15
\][/tex]
Therefore, the value of [tex]\(b\)[/tex] for the binomial expansion is [tex]\(15\)[/tex].
1. Identify the Coefficient of the Highest Degree Term:
The highest degree term is [tex]\(625x^4\)[/tex]. Since the polynomial is in the form [tex]\((ax+b)^4\)[/tex], the coefficient of [tex]\(x^4\)[/tex] should be [tex]\(a^4\)[/tex]. Therefore:
[tex]\[
a^4 = 625
\][/tex]
To find [tex]\(a\)[/tex], we need to take the fourth root of 625:
[tex]\[
\sqrt[4]{625} = 5
\][/tex]
Thus, [tex]\(a = 5\)[/tex].
2. Determine the Constant Term:
The constant term of the expansion [tex]\((ax+b)^4\)[/tex] is [tex]\(b^4\)[/tex]. In the polynomial, this constant term is 50,625:
[tex]\[
b^4 = 50,625
\][/tex]
To find [tex]\(b\)[/tex], take the fourth root of 50,625:
[tex]\[
b = \sqrt[4]{50,625} = 15
\][/tex]
Therefore, the value of [tex]\(b\)[/tex] for the binomial expansion is [tex]\(15\)[/tex].
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