## What is factor theorem with examples?

Answer: An example of factor theorem can be the factorization of 6×2 + 17x + 5 by splitting the middle term. In this example, one can find two numbers, ‘p’ and ‘q’ in a way such that, p + q = 17 and pq = 6 x 5 = 30. After that one can get the factors.

**What is the factor theorem for polynomials?**

Factor theorem is usually used to factor and find the roots of polynomials. A root or zero is where the polynomial is equal to zero. Therefore, the theorem simply states that when f(k) = 0, then (x – k) is a factor of f(x).

**What is an example of a polynomial in factored form?**

If a polynomial of lowest degree p has zeros at x=x1,x2,…,xn x = x 1 , x 2 , … , x n , then the polynomial can be written in the factored form: f(x)=a(x−x1)p1(x−x2)p2⋯(x−xn)pn f ( x ) = a ( x − x 1 ) p 1 ( x − x 2 ) p 2 ⋯ ( x − x n ) p n where the powers pi on each factor can be determined by the behavior of the graph …

### What is factor theorem Class 9 examples?

According to factor theorem, if f(x) is a polynomial of degree n ≥ 1 and ‘a’ is any real number, then, (x-a) is a factor of f(x), if f(a)=0. Also, we can say, if (x-a) is a factor of polynomial f(x), then f(a) = 0. This proves the converse of the theorem. Let us see the proof of this theorem along with examples.

**What is the factor theorem of Class 9?**

The Factor theorem is a unique case consideration of the polynomial remainder theorem. Thus the factor theorem states that a polynomial has a factor if and only if: The polynomial x – M is a factor of the polynomial f(x) if and only if f (M) = 0.

**What does the factor theorem tell?**

In practice, the Factor Theorem is used when factoring polynomials “completely”. Any time you divide by a number (being a potential root of the polynomial) and get a zero remainder in the synthetic division, this means that the number is indeed a root, and thus “x minus the number” is a factor.

#### What is the proof of factor theorem?

The proof of The Factor Theorem is a consequence of what we already know. If (x−c) is a factor of p(x), this means p(x)=(x−c)q(x) for some polynomial q. Hence, p(c)=(c−c)q(c)=0, so c is a zero of p. Conversely, if c is a zero of p, then p(c)=0.

**What is the standard form of polynomial function?**

A polynomial is a function of the form f(x) = anxn + an−1xn−1 + + a2x2 + a1x + a0 . The degree of a polynomial is the highest power of x in its expression. Constant (non-zero) polynomials, linear polynomials, quadratics, cubics and quartics are polynomials of degree 0, 1, 2 , 3 and 4 respectively.

**Is factor theorem and remainder theorem same?**

The remainder theorem tells us that for any polynomial f(x) , if you divide it by the binomial x−a , the remainder is equal to the value of f(a) . The factor theorem tells us that if a is a zero of a polynomial f(x) , then (x−a) is a factor of f(x) , and vice-versa.

## How do you find the factor theorem of Class 9?

CBSE NCERT Notes Class 9 Maths Polynomials. x – a is a factor of the polynomial p(x), if p(a) = 0. Also, if x – a is a factor of p(x), then p(a) = 0, where a is any real number. This is an extension to remainder theorem where remainder is 0, i.e. p(a) = 0.

**Which is an example of the factor theorem?**

As discussed in the introduction, a polynomial f (x) has a factor (x-a), if and only if, f (a) = 0. It is one of the methods to do the factorisation of a polynomial. Here we will prove the factor theorem, according to which we can factorise the polynomial. Consider a polynomial f (x) which is divided by (x-c), then f (c)=0.

**How to find the factors of a polynomial?**

The steps are given below to find the factors of a polynomial using factor theorem: Step 1 : If f (-c)=0, then (x+ c) is a factor of the polynomial f (x). Step 2 : If p (d/c)= 0, then (cx-d) is a factor of the polynomial f (x). Step 3 : If p (-d/c)= 0, then (cx+d) is a factor of the polynomial f (x).

### When do you call a polynomial a depressed polynomial?

There is another way to define the factor theorem. Usually, when a polynomial is divided by a binomial, we will get a reminder. The quotient obtained is called as depressed polynomial when the polynomial is divided by one of its binomial factors. If you get the remainder as zero, the factor theorem is illustrated as follows:

**What is the purpose of factor theorem class 9?**

Go through once and get a clear understanding of this theorem. Factor theorem class 9 maths polynomial enables the children to get a knowledge of finding the roots of quadratic expressions and the polynomial equations, which is used for solving complex problems in your higher studies.