When two rational expressions are divide, the quotient is 5x² + 3x + 12 + 7/(x + 5. What are the two polynomials?
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Work backwards by finding a common denominator ...
5x² + 3x + 12 + 7/(x + 5)
= [5x^2(x+5) + 3x(x+5) + 12(x+5) + 7]/(x+5)
= (5x^3 + 25x^2 + 3x^2 + 15x + 12x + 60 + 7)/(x+5)
= (5x^3 + 28x^2 + 27x + 67)/(x+5)
The expression above is a rational expression, not a quotient.
A quotient is just a polynomial, and does not include the division.
Take another look at the problem and post it as it says.
To find the two polynomials, we need to divide the given rational expression by 5x^2 + 3x + 12. Let's break it down step-by-step:
Step 1: Write the given rational expression:
(Rational expression) / (Divisor) = Quotient
Step 2: Rewrite the divisor in factored form:
5x^2 + 3x + 12 = (x + 5)(5x + 2)
Step 3: Rewrite the quotient as "0" + 7/(x + 5):
Quotient = 5x^2 + 3x + 12 + 7/(x + 5)
Step 4: Divide the rational expression by the divisor:
(Rational expression) / (Divisor) = (Quotient) = 0 + 7/(x + 5)
Therefore, the two polynomials are:
Rational expression = 0
Divisor = 5x^2 + 3x + 12
To find the two polynomials, we first need to rewrite the given quotient as a division of two rational expressions. The given quotient can be written as:
5x² + 3x + 12 + 7/(x + 5)
To express this as a division of two rational expressions, we can rewrite the polynomial part as a fraction with a denominator of 1. This gives us:
(5x² + 3x + 12(x + 5) + 7)/(x + 5)
Now, we can determine the two polynomials:
1. The numerator of the rational expression: 5x² + 3x + 12(x + 5) + 7
2. The denominator of the rational expression: x + 5
Therefore, the two polynomials are:
1. Numerator polynomial: 5x² + 3x + 12(x + 5) + 7
2. Denominator polynomial: x + 5