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Multiply the rational expression and write the product in simplest form: x2+2x−3x2+3x−4⋅x2+6x+8x2−9 .(1 point)

Responses

−(x+2)x+3
Start Fraction negative left parenthesis x plus 2 right parenthesis over x plus 3 End Fraction

(x+4)(x+2)(x+3)(x−3)
Start Fraction left parenthesis x plus 4 right parenthesis left parenthesis x plus 2 right parenthesis over left parenthesis x plus 3 right parenthesis left parenthesis x minus 3 right parenthesis End Fraction

x+2x−3
Start Fraction x plus 2 over x minus 3 End Fraction

−23

7 answers

To multiply the rational expression \(\frac{x^2 + 2x - 3}{x^2 + 3x - 4} \cdot \frac{x^2 + 6x + 8}{x^2 - 9}\), we need to first factor each of the polynomials.

  1. Factor the first numerator \(x^2 + 2x - 3\): \[ x^2 + 2x - 3 = (x + 3)(x - 1) \]

  2. Factor the first denominator \(x^2 + 3x - 4\): \[ x^2 + 3x - 4 = (x + 4)(x - 1) \]

  3. Factor the second numerator \(x^2 + 6x + 8\): \[ x^2 + 6x + 8 = (x + 2)(x + 4) \]

  4. Factor the second denominator \(x^2 - 9\) (difference of squares): \[ x^2 - 9 = (x - 3)(x + 3) \]

Now substitute the factored forms back into the expression:

\[ \frac{(x + 3)(x - 1)}{(x + 4)(x - 1)} \cdot \frac{(x + 2)(x + 4)}{(x - 3)(x + 3)} \]

Next, we can combine the fractions:

\[ \frac{(x + 3)(x - 1)(x + 2)(x + 4)}{(x + 4)(x - 1)(x - 3)(x + 3)} \]

Now, we can cancel the common factors in the numerator and denominator. Notice that \((x + 3)\) and \((x - 1)\) appear in both:

  • Cancel \((x + 3)\)
  • Cancel \((x - 1)\)
  • Cancel \((x + 4)\)

This simplifies our expression to:

\[ \frac{(x + 2)}{(x - 3)} \]

Thus, the final simplified form of the product is:

\[ \frac{x + 2}{x - 3} \]

The correct response is: \[ \text{Start Fraction } x + 2 \text{ over } x - 3 \text{ End Fraction} \]

Write the polynomial expression in simplest form: x2−9x2−4⋅4x−812x+36 .(1 point)
Responses

13
Start Fraction 1 over 3 End Fraction

x−33(x+2)
Start Fraction x minus 3 over 3 left parenthesis x plus 2 right parenthesis End Fraction

(x−3)(x−2)3(x+2)(x+2)
Start Fraction left parenthesis x minus 3 right parenthesis left parenthesis x minus 2 right parenthesis over 3 left parenthesis x plus 2 right parenthesis left parenthesis x plus 2 right parenthesis End Fraction

2(x−3)6(x+2)

To simplify the expression \(\frac{x^2 - 9}{x^2 - 4} \cdot \frac{4x - 8}{12x + 36}\), we first need to factor each polynomial component in the expression.

  1. Factor the first numerator \(x^2 - 9\) (difference of squares): \[ x^2 - 9 = (x - 3)(x + 3) \]

  2. Factor the first denominator \(x^2 - 4\) (difference of squares): \[ x^2 - 4 = (x - 2)(x + 2) \]

  3. Factor the second numerator \(4x - 8\): \[ 4x - 8 = 4(x - 2) \]

  4. Factor the second denominator \(12x + 36\): \[ 12x + 36 = 12(x + 3) \]

Now we can substitute these factored forms back into the expression:

\[ \frac{(x - 3)(x + 3)}{(x - 2)(x + 2)} \cdot \frac{4(x - 2)}{12(x + 3)} \]

Next, we combine these fractions:

\[ \frac{(x - 3)(x + 3) \cdot 4(x - 2)}{(x - 2)(x + 2) \cdot 12(x + 3)} \]

Now we can simplify by canceling out common factors in the numerator and denominator.

  • Cancel \((x - 2)\)
  • Cancel \((x + 3)\)

This simplifies to:

\[ \frac{4(x - 3)}{12(x + 2)} \]

Now we can simplify:

\[ = \frac{4}{12} \cdot \frac{(x - 3)}{(x + 2)} = \frac{1}{3} \cdot \frac{(x - 3)}{(x + 2)} = \frac{x - 3}{3(x + 2)} \]

Thus, the final simplified form of the expression is:

\[ \frac{x - 3}{3(x + 2)} \]

The correct response is: \[ \text{Start Fraction } x - 3 \text{ over } 3(x + 2) \text{ End Fraction} \]

Express in simplest form: x2+8x−48x2−144 ÷(4−x) (1 point)
Responses

−1x−12
the fraction with numerator negative 1 and denominator x minus 12

1x−12
the fraction with numerator 1 and denominator x minus 12

−(x+4)(x−4)(x+12)
the fraction with numerator negative open paren x plus 4 close paren and denominator open paren x minus 4 close paren times open paren x plus 12 close paren

(x−4)(x+4)x−12

To simplify the expression \(\frac{x^2 + 8x - 48}{x^2 - 144} \div (4 - x)\), we first rewrite the division as multiplication by the reciprocal:

\[ \frac{x^2 + 8x - 48}{x^2 - 144} \cdot \frac{1}{4 - x} \]

Next, we will factor the numerator and denominator.

  1. Factor the numerator \(x^2 + 8x - 48\): To factor \(x^2 + 8x - 48\), we need two numbers that multiply to \(-48\) and add to \(8\). The numbers are \(12\) and \(-4\). Thus: \[ x^2 + 8x - 48 = (x + 12)(x - 4) \]

  2. Factor the denominator \(x^2 - 144\) (difference of squares): \[ x^2 - 144 = (x - 12)(x + 12) \]

Now, substituting these factored forms back into the expression:

\[ \frac{(x + 12)(x - 4)}{(x - 12)(x + 12)} \cdot \frac{1}{4 - x} \]

Next, recognize that \(4 - x = -(x - 4)\). Thus, we can rewrite the expression as follows:

\[ \frac{(x + 12)(x - 4)}{(x - 12)(x + 12)} \cdot \frac{-1}{(x - 4)} \]

Now we multiply:

\[ = \frac{-(x + 12)(x - 4)}{(x - 12)(x + 12)(x - 4)} \]

Now we can cancel the common factor \((x - 4)\) from the numerator and denominator:

\[ = \frac{-(x + 12)}{(x - 12)} \]

This simplifies to:

\[ = -\frac{x + 12}{x - 12} \]

Thus, the final expression in simplest form is:

\[ -\frac{1 \cdot (x + 12)}{x - 12} \]

The answer corresponds to the option: \[ -\frac{1}{x - 12} \text{ (the fraction with numerator negative 1 and denominator } x - 12) \]

Express in simplest form: 3x+184x+8÷x2−36x+2 .(1 point)
Responses

4(x−6)3
Start Fraction 4 left parenthesis x minus 6 right parenthesis over 3 End Fraction

34x−8
Start Fraction 3 over 4 x minus 8 End Fraction

34x−24
Start Fraction 2 over 4 x minus 24 End Fraction

14(x−2)

To simplify the expression \( \frac{3x + 18}{4x + 8} \div (x^2 - 36x + 2) \), we will first convert the division into multiplication by taking the reciprocal of the second term, followed by factoring where possible.

  1. Rewrite the division:

\[ \frac{3x + 18}{4x + 8} \cdot \frac{1}{x^2 - 36x + 2} \]

  1. Factor the numerator \(3x + 18\):

\[ 3x + 18 = 3(x + 6) \]

  1. Factor the denominator \(4x + 8\):

\[ 4x + 8 = 4(x + 2) \]

  1. Now we need to factor the expression \(x^2 - 36x + 2\), but it does not factor easily into integers. We can use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):

For \(a = 1\), \(b = -36\), and \(c = 2\):

\[ x = \frac{36 \pm \sqrt{(-36)^2 - 4 \cdot 1 \cdot 2}}{2 \cdot 1} \]

\[ = \frac{36 \pm \sqrt{1296 - 8}}{2} = \frac{36 \pm \sqrt{1288}}{2} = \frac{36 \pm 2\sqrt{322}}{2} = 18 \pm \sqrt{322} \]

Since \(x^2 - 36x + 2\) does not factor into nice integers, we will proceed without factoring it explicitly. Using its roots won't bear much simplification in this case.

  1. Putting it all together:

Now substituting our factored forms into the original expression gives us:

\[ \frac{3(x + 6)}{4(x + 2)} \cdot \frac{1}{x^2 - 36x + 2} \]

This gives us:

\[ \frac{3(x + 6)}{4(x + 2)(x^2 - 36x + 2)} \]

Since we can't simplify \(x^2 - 36x + 2\) further with the current terms, we'll report that.

To represent this according to your provided choices:

  • The expressions given don't quite match since we dealt with quadratic factors and the simplifications detailed. The expected responses may need proper identification as the above is more complex involving roots for further calculations.

Thus, our final answer will be:

\[ \frac{3(x + 6)}{4(x + 2)(x^2 - 36x + 2)} \]

Check your choices again, as our expression is not among your listed options based on my current factoring process.