question 1: build the fraction x+5/x+3 to an equivalent fraction whoes denominator is x^2
question 2: simplify (x^2+6x+9)(3-x)/(x-3)(2x+6)
question 3. find the product in simplest from x^3-9x/x^5+27x^2 * x^2-3x+9/x^2+3x
First, observe the rule of operator precedence, i.e. BEDMAS (brackets,exponents, division/multiplication, addition/subtration) in that order.
Q1.
I suspect the expression should have been written as (x+5)/(x+3) to be interpreted correctly.
Based on that, we know that the new denominator is x², and denote the new numerator as N, then equate the two equivalent fractions:
(x+5)/(x+3) = N / x²
Cross multiply and solve for N:
N = (x+5)*x²/(x+3)
=(x³+4x²)/(x+3)
Unfortunately N is not a polynomial.
For Q2 and Q3, I suggest you correct the expressions according to the BEDMAS precedence rules and repost.
typo:
N = (x+5)*x²/(x+3)
=(x³+5x²)/(x+3)
To solve these questions, let's break them down step by step:
Question 1: Building the fraction (x+5)/(x+3) to an equivalent fraction with a denominator of x^2.
To build an equivalent fraction, we need to multiply both the numerator and denominator by the same expression that will result in the desired denominator. In this case, we want the denominator to be x^2.
Step 1: Multiply the numerator and denominator by (x-3), which will cancel out the (x+3) in the denominator:
[(x+5)(x-3)] / [(x+3)(x-3)]
Step 2: Simplify the numerator and denominator:
(x^2 + 2x - 15) / (x^2 - 9)
Therefore, the equivalent fraction of (x+5)/(x+3) with a denominator of x^2 is (x^2 + 2x - 15)/(x^2 - 9).
Question 2: Simplifying (x^2+6x+9)(3-x)/(x-3)(2x+6)
To simplify this expression, we can factorize and cancel out common factors if possible.
Step 1: Factorize the numerator and denominator:
[(x+3)(x+3)(3-x)] / [(x-3)(2)(x+3)]
Step 2: Cancel out common factors between the numerator and the denominator:
[(x+3)(3-x)] / [(x-3)(2)]
Step 3: Simplify the expression in the numerator:
[(3-x)(x+3)] / [(x-3)(2)]
Step 4: Rearrange the terms:
[-(x-3)(x+3)] / [(x-3)(2)]
Step 5: Cancel out the common factor (x-3):
-(x+3) / 2
Therefore, the simplified form of (x^2+6x+9)(3-x)/(x-3)(2x+6) is -(x+3)/2.
Question 3: Finding the product (x^3-9x)/(x^5+27x^2) * (x^2-3x+9)/(x^2+3x).
To find the product, multiply the numerators together and multiply the denominators together.
Step 1: Multiply the numerators:
(x^3-9x)(x^2-3x+9)
Step 2: Multiply the denominators:
(x^5+27x^2)(x^2+3x)
Step 3: Simplify the numerator and denominator separately:
Numerator: x^5 - 3x^4 + 9x^3 - 9x^3 + 27x^2 - 81x
Denominator: x^7 + 3x^6 + 3x^4 + 9x^3 + 27x^2 + 81x
Step 4: Combine like terms in the numerator and denominator:
Numerator: x^5 - 3x^4 + 27x^2 - 81x
Denominator: x^7 + 3x^6 + 3x^4 + 9x^3 + 27x^2 + 81x
Step 5: There are no common factors that can be canceled out, so the product remains as:
(x^5 - 3x^4 + 27x^2 - 81x) / (x^7 + 3x^6 + 3x^4 + 9x^3 + 27x^2 + 81x)
Therefore, the product in simplified form is (x^5 - 3x^4 + 27x^2 - 81x) / (x^7 + 3x^6 + 3x^4 + 9x^3 + 27x^2 + 81x).