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If the value of the polynomial \(p(x)\) at \(x = k\) is zero \(p(k) = 0\), then the real number \(k\) is called the zero of the polynomial \(p(x)\).

Important!

To find the zeroes of a quadratic polynomial, put \(p(x) = 0\).

Example:

**1**. Find the zeroes of the quadratic expression \(p(x) = 2x^2 + 3x + 1\).

**Solution**:

\(p(x) = 2x^2 + 3x + 1\)

Put \(p(x) = 0\).

\(2x^2 + 3x + 1 = 0\)

\(2x^2 + 2x + x + 1 = 0\)

\(2x(x + 1) + 1(x + 1) = 0\)

\((2x + 1) (x + 1) = 0\)

\(2x + 1 = 0\) or \(x + 1 = 0\)

\(x = \frac{-1}{2}\) or \(x = -1\)

**Thus, the zeroes of**\(p(x) = 2x^2 + 3x + 1\)

**are**\(\frac{-1}{2}\)

**and**\(-1\).

**2**. Check \(2\) is a zero of the polynomial \(p(x) = 3x^2 - 5x - 2\).

**Solution**:

\(p(x) = 3x^2 - 5x - 2\)

Put \(x = 2\) in \(p(x)\).

\(p(2) = 3(2)^2 - 5(2) - 2\)

\(= 3(4) - 10 - 2\)

\(= 12 - 12\)

\(= 0 \)

**Thus**, \(2\)

**is a zero of the polynomial**\(p(x) = 3x^2 - 5x - 2\).