What Value Of $c$ Makes The Polynomial Below A Perfect Square?$x^2 + 10x + C$$c =$A. 20 B. 5 C. 25 D. 100

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What Value of c Makes the Polynomial a Perfect Square?

In algebra, a perfect square trinomial is a polynomial that can be factored into the square of a binomial. A perfect square trinomial has the form of a2+2ab+b2{a^2 + 2ab + b^2} or a2−2ab+b2{a^2 - 2ab + b^2}, where a{a} and b{b} are constants. In this article, we will explore the value of c{c} that makes the polynomial x2+10x+c{x^2 + 10x + c} a perfect square.

Understanding Perfect Square Trinomials

A perfect square trinomial can be factored into the square of a binomial using the formula (a+b)2=a2+2ab+b2{(a + b)^2 = a^2 + 2ab + b^2} or (a−b)2=a2−2ab+b2{(a - b)^2 = a^2 - 2ab + b^2}. To determine if a trinomial is a perfect square, we need to check if it can be factored into the square of a binomial.

The Given Polynomial

The given polynomial is x2+10x+c{x^2 + 10x + c}. To make this polynomial a perfect square, we need to find the value of c{c} that allows it to be factored into the square of a binomial.

Factoring the Polynomial

Let's assume that the polynomial can be factored into the square of a binomial: (x+a)2=x2+2ax+a2{(x + a)^2 = x^2 + 2ax + a^2}. Comparing this with the given polynomial, we can see that 2ax=10x{2ax = 10x}, which implies that a=5{a = 5}.

Finding the Value of c

Now that we have found the value of a{a}, we can substitute it into the formula for the square of a binomial: (x+5)2=x2+10x+25{(x + 5)^2 = x^2 + 10x + 25}. Comparing this with the given polynomial, we can see that c=25{c = 25}.

In conclusion, the value of c{c} that makes the polynomial x2+10x+c{x^2 + 10x + c} a perfect square is 25.

The correct answer is C. 25.

Here are a few additional examples of perfect square trinomials:

  • x2+6x+9=(x+3)2{x^2 + 6x + 9 = (x + 3)^2}
  • x2−4x+4=(x−2)2{x^2 - 4x + 4 = (x - 2)^2}
  • x2+2x+1=(x+1)2{x^2 + 2x + 1 = (x + 1)^2}

These examples demonstrate how to factor perfect square trinomials into the square of a binomial.

Here are a few tips and tricks for factoring perfect square trinomials:

  • Look for two perfect squares that add up to the constant term.
  • Look for two perfect squares that subtract to the constant term.
  • Use the formula (a+b)2=a2+2ab+b2{(a + b)^2 = a^2 + 2ab + b^2} or (a−b)2=a2−2ab+b2{(a - b)^2 = a^2 - 2ab + b^2} to factor the trinomial.

By following these tips and tricks, you can easily factor perfect square trinomials into the square of a binomial.

Here are a few common mistakes to avoid when factoring perfect square trinomials:

  • Not looking for two perfect squares that add up to the constant term.
  • Not looking for two perfect squares that subtract to the constant term.
  • Not using the formula (a+b)2=a2+2ab+b2{(a + b)^2 = a^2 + 2ab + b^2} or (a−b)2=a2−2ab+b2{(a - b)^2 = a^2 - 2ab + b^2} to factor the trinomial.

By avoiding these common mistakes, you can ensure that you are factoring perfect square trinomials correctly.

Here are some frequently asked questions about perfect square trinomials:

Q: What is a perfect square trinomial?

A: A perfect square trinomial is a polynomial that can be factored into the square of a binomial. It has the form of a2+2ab+b2{a^2 + 2ab + b^2} or a2−2ab+b2{a^2 - 2ab + b^2}, where a{a} and b{b} are constants.

Q: How do I determine if a trinomial is a perfect square?

A: To determine if a trinomial is a perfect square, you need to check if it can be factored into the square of a binomial. You can use the formula (a+b)2=a2+2ab+b2{(a + b)^2 = a^2 + 2ab + b^2} or (a−b)2=a2−2ab+b2{(a - b)^2 = a^2 - 2ab + b^2} to factor the trinomial.

Q: What is the formula for factoring a perfect square trinomial?

A: The formula for factoring a perfect square trinomial is (a+b)2=a2+2ab+b2{(a + b)^2 = a^2 + 2ab + b^2} or (a−b)2=a2−2ab+b2{(a - b)^2 = a^2 - 2ab + b^2}.

Q: How do I find the value of c in a perfect square trinomial?

A: To find the value of c{c} in a perfect square trinomial, you need to factor the trinomial into the square of a binomial. You can use the formula (a+b)2=a2+2ab+b2{(a + b)^2 = a^2 + 2ab + b^2} or (a−b)2=a2−2ab+b2{(a - b)^2 = a^2 - 2ab + b^2} to factor the trinomial.

Q: What are some common mistakes to avoid when factoring perfect square trinomials?

A: Some common mistakes to avoid when factoring perfect square trinomials include:

  • Not looking for two perfect squares that add up to the constant term.
  • Not looking for two perfect squares that subtract to the constant term.
  • Not using the formula (a+b)2=a2+2ab+b2{(a + b)^2 = a^2 + 2ab + b^2} or (a−b)2=a2−2ab+b2{(a - b)^2 = a^2 - 2ab + b^2} to factor the trinomial.

Q: How do I apply the formula for factoring perfect square trinomials?

A: To apply the formula for factoring perfect square trinomials, you need to:

  1. Identify the constant term in the trinomial.
  2. Look for two perfect squares that add up to the constant term or subtract to the constant term.
  3. Use the formula (a+b)2=a2+2ab+b2{(a + b)^2 = a^2 + 2ab + b^2} or (a−b)2=a2−2ab+b2{(a - b)^2 = a^2 - 2ab + b^2} to factor the trinomial.

Q: What are some examples of perfect square trinomials?

A: Some examples of perfect square trinomials include:

  • x2+6x+9=(x+3)2{x^2 + 6x + 9 = (x + 3)^2}
  • x2−4x+4=(x−2)2{x^2 - 4x + 4 = (x - 2)^2}
  • x2+2x+1=(x+1)2{x^2 + 2x + 1 = (x + 1)^2}

Q: How do I check if a trinomial is a perfect square?

A: To check if a trinomial is a perfect square, you need to factor the trinomial into the square of a binomial. You can use the formula (a+b)2=a2+2ab+b2{(a + b)^2 = a^2 + 2ab + b^2} or (a−b)2=a2−2ab+b2{(a - b)^2 = a^2 - 2ab + b^2} to factor the trinomial.

In conclusion, perfect square trinomials are polynomials that can be factored into the square of a binomial. By following the tips and tricks outlined in this article, you can easily factor perfect square trinomials into the square of a binomial.