What Is The Factored Form Of The Polynomial $z^2 - 10z + 25$?A. $(z - 5)(z - 5)$ B. \$(z + 5)(z + 5)$[/tex\] C. $(z - 2)(z + 5)$ D. $(z + 2)(z - 5)$

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Introduction to Factoring Polynomials

Factoring polynomials is a fundamental concept in algebra that involves expressing a polynomial as a product of simpler polynomials. This process is essential in solving equations, finding roots, and simplifying expressions. In this article, we will focus on factoring the polynomial z2−10z+25z^2 - 10z + 25 and explore the different methods used to achieve this.

Understanding the Polynomial

The given polynomial is a quadratic expression in the form of az2+bz+caz^2 + bz + c. In this case, a=1a = 1, b=−10b = -10, and c=25c = 25. To factor this polynomial, we need to find two binomials whose product equals the original polynomial.

Factoring by Grouping

One method of factoring is by grouping. This involves factoring the polynomial into two binomials by grouping the terms. However, in this case, the polynomial does not lend itself easily to factoring by grouping.

Factoring by Completing the Square

Another method of factoring is by completing the square. This involves manipulating the polynomial to create a perfect square trinomial. To do this, we need to add and subtract a constant term to the polynomial.

Factoring the Polynomial

To factor the polynomial z2−10z+25z^2 - 10z + 25, we can use the method of completing the square. We start by adding and subtracting (10/2)2=25(10/2)^2 = 25 to the polynomial:

z2−10z+25=(z2−10z+25)+25−25z^2 - 10z + 25 = (z^2 - 10z + 25) + 25 - 25

This can be rewritten as:

(z2−10z+25)+25−25=(z2−10z+25)+0(z^2 - 10z + 25) + 25 - 25 = (z^2 - 10z + 25) + 0

Now, we can factor the perfect square trinomial:

(z2−10z+25)+0=(z−5)2(z^2 - 10z + 25) + 0 = (z - 5)^2

Identifying the Factored Form

The factored form of the polynomial z2−10z+25z^2 - 10z + 25 is (z−5)2(z - 5)^2. This can be rewritten as:

(z−5)(z−5)(z - 5)(z - 5)

Conclusion

In conclusion, the factored form of the polynomial z2−10z+25z^2 - 10z + 25 is (z−5)2(z - 5)^2 or (z−5)(z−5)(z - 5)(z - 5). This is the correct answer among the options provided.

Final Answer

The final answer is:

A. $(z - 5)(z - 5)$

Introduction

Factoring polynomials is a fundamental concept in algebra that involves expressing a polynomial as a product of simpler polynomials. In our previous article, we explored the factored form of the polynomial z2−10z+25z^2 - 10z + 25. In this article, we will answer some frequently asked questions about factoring polynomials.

Q: What is the difference between factoring and simplifying a polynomial?

A: Factoring a polynomial involves expressing it as a product of simpler polynomials, while simplifying a polynomial involves combining like terms to reduce the polynomial to its simplest form.

Q: How do I know if a polynomial can be factored?

A: A polynomial can be factored if it can be expressed as a product of simpler polynomials. This can be determined by looking for common factors among the terms or by using methods such as factoring by grouping or completing the square.

Q: What is the difference between factoring by grouping and factoring by completing the square?

A: Factoring by grouping involves factoring a polynomial by grouping the terms into two or more groups and factoring each group separately. Factoring by completing the square involves manipulating the polynomial to create a perfect square trinomial.

Q: How do I factor a polynomial with a negative leading coefficient?

A: To factor a polynomial with a negative leading coefficient, you can multiply the entire polynomial by -1 to make the leading coefficient positive. Then, you can factor the polynomial using the methods mentioned earlier.

Q: Can a polynomial with a degree greater than 2 be factored?

A: Yes, a polynomial with a degree greater than 2 can be factored, but it may not be possible to factor it into linear factors. In such cases, you may need to use more advanced techniques such as the rational root theorem or synthetic division.

Q: How do I determine the number of factors of a polynomial?

A: The number of factors of a polynomial can be determined by looking at the degree of the polynomial. A polynomial of degree n can have up to n factors.

Q: Can a polynomial be factored into a product of two binomials?

A: Yes, a polynomial can be factored into a product of two binomials if it can be expressed as a product of two simpler polynomials.

Q: How do I check if a factored form of a polynomial is correct?

A: To check if a factored form of a polynomial is correct, you can multiply the factors together and simplify the expression. If the result is the original polynomial, then the factored form is correct.

Q: Can a polynomial be factored into a product of three or more binomials?

A: Yes, a polynomial can be factored into a product of three or more binomials if it can be expressed as a product of three or more simpler polynomials.

Conclusion

In conclusion, factoring polynomials is a fundamental concept in algebra that involves expressing a polynomial as a product of simpler polynomials. By understanding the different methods of factoring and the properties of polynomials, you can master the art of factoring and simplify complex expressions.

Final Answer

The final answer is:

  • Factoring a polynomial involves expressing it as a product of simpler polynomials.
  • A polynomial can be factored if it can be expressed as a product of simpler polynomials.
  • Factoring by grouping and factoring by completing the square are two methods of factoring polynomials.
  • A polynomial with a degree greater than 2 can be factored, but it may not be possible to factor it into linear factors.
  • The number of factors of a polynomial can be determined by looking at the degree of the polynomial.
  • A polynomial can be factored into a product of two binomials if it can be expressed as a product of two simpler polynomials.
  • To check if a factored form of a polynomial is correct, you can multiply the factors together and simplify the expression.