Zeros of Polynomials

The zero of a polynomial is a value of the variable that makes the polynomial equal to zero. In simple words, if substituting a number into a polynomial gives the result 0, that number is called a zero (or root) of the polynomial.

Zeros help us understand how the polynomial behaves and where its graph touches or cuts the x-axis.

Example: For the polynomial \(p(x) = x - 5\), the zero is \(x = 5\) because \(p(5) = 0\).

A linear polynomial has the form \(p(x) = ax + b\), where \(a \neq 0\). It has exactly one zero.

To find the zero, set the polynomial equal to 0 and solve:

\(ax + b = 0\)

\(x = -\dfrac{b}{a}\)

A quadratic polynomial has the form \(p(x) = ax^2 + bx + c\). It may have 0, 1, or 2 zeros. At this stage, you only need to verify whether a given number is a zero.

To verify, substitute the number and check if the value becomes zero.

The zero of a polynomial corresponds to the value of \(x\) where the graph of the polynomial touches or cuts the x-axis.

For polynomial \(p(x)\):

• If the graph cuts the x-axis → real zeros exist.
• If the graph touches the x-axis → repeated zero.
• If the graph does not touch the x-axis → no real zero.

If \(x = a\) is a zero of polynomial \(p(x)\), the graph passes through the point \((a, 0)\).

• Thinking the value that makes polynomial equal to 1 is a zero (it must equal 0).
• Confusing zeros with factors.
• Not substituting correctly while verifying zeros.
• Assuming every polynomial must have a zero (some do not).

Find or verify the zeros:

1. Find zero of \(p(x) = 4x - 8\)
2. Verify if 2 is a zero of \(p(x) = x^2 - 4\)
3. Verify if −3 is a zero of \(p(x) = x^2 + 5x + 6\)
4. Find the zero of \(p(x) = -5x + 20\)

• Zeros of a polynomial are the values of variable for which the polynomial becomes 0.
• Linear polynomials have exactly one zero.
• Quadratic polynomials may have 0, 1, or 2 zeros.
• Zeros correspond to points where the graph touches/cuts the x-axis.
• Zeros are checked by direct substitution.