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Introduction to Linear Polynomials - Visualising linear relationships

Grade 9CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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A polynomial of degree 11 is called a linear polynomial. Its general form is p(x)=ax+bp(x) = ax + b, where aa and bb are real numbers and a≠0a \neq 0.

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The value of xx for which p(x)=0p(x) = 0 is called the zero of the linear polynomial. For p(x)=ax+bp(x) = ax + b, the zero is x=−bax = -\frac{b}{a}.

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Geometrically, the graph of a linear equation y=ax+by = ax + b is always a straight line.

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The zero of a linear polynomial p(x)p(x) is the xx-coordinate of the point where the graph of y=p(x)y = p(x) intersects the xx-axis.

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A linear polynomial has exactly one zero.

📐Formulae

p(x)=ax+b, where a≠0p(x) = ax + b, \text{ where } a \neq 0

Zero of p(x)=−Constant termCoefficient of x=−ba\text{Zero of } p(x) = -\frac{\text{Constant term}}{\text{Coefficient of } x} = -\frac{b}{a}

y=mx+cy = mx + c

💡Examples

Problem 1:

Find the zero of the linear polynomial p(x)=2x+8p(x) = 2x + 8.

Solution:

To find the zero, set p(x)=0p(x) = 0: 2x+8=02x + 8 = 0 2x=−82x = -8 x=−82x = \frac{-8}{2} x=−4x = -4

Explanation:

The zero of the polynomial is the value of xx that makes the expression equal to zero. Here, substituting x=−4x = -4 gives 2(−4)+8=02(-4) + 8 = 0.

Problem 2:

Verify if x=32x = \frac{3}{2} is a zero of the polynomial p(x)=4x−6p(x) = 4x - 6.

Solution:

Substitute x=32x = \frac{3}{2} into p(x)p(x): p(32)=4(32)−6p\left(\frac{3}{2}\right) = 4\left(\frac{3}{2}\right) - 6 p(32)=2×3−6p\left(\frac{3}{2}\right) = 2 \times 3 - 6 p(32)=6−6=0p\left(\frac{3}{2}\right) = 6 - 6 = 0 Since p(32)=0p\left(\frac{3}{2}\right) = 0, x=32x = \frac{3}{2} is a zero.

Explanation:

If substituting a value into the polynomial results in zero, that value is a root or zero of the polynomial.

Problem 3:

Determine the point where the graph of y=3x−9y = 3x - 9 intersects the xx-axis.

Solution:

The graph intersects the xx-axis when y=0y = 0: 0=3x−90 = 3x - 9 3x=93x = 9 x=3x = 3 The point of intersection is (3,0)(3, 0).

Explanation:

The intersection with the xx-axis corresponds to the zero of the linear polynomial. The coordinates are always in the form (zero,0)(\text{zero}, 0).