Form Quadratic Polynomial from Sum and Product of Zeroes

 

❓ Question

Find a quadratic polynomial for each pair where the given numbers are respectively the:

• Sum of zeroes (S)
• Product of zeroes (P)

(i) 1/4, -1
(ii) √2, 1/3
(iii) 0, √5
(iv) 1, 1
(v) -1/4, 1/4
(vi) 4, 1

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🖼️ Solution Image

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✍️ Short Concept

For a quadratic polynomial:

$$ax^2+bx+c$$

if:

$$\text{Sum of zeroes}=S$$

and

$$\text{Product of zeroes}=P$$

then the polynomial is:

$$f(x)=x^2-Sx+P$$

💯

If fractions are present, we multiply throughout to remove denominators.

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🔷 (i) $S=\frac14,\ P=-1$

Using:

$$f(x)=x^2-\frac14x-1$$

Multiplying by 4:

$$\boxed{4x^2-x-4}$$

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🔷 (ii) $S=\sqrt2,\ P=\frac13$

$$f(x)=x^2-\sqrt2x+\frac13$$

Multiplying by 3:

$$\boxed{3x^2-3\sqrt2x+1}$$

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🔷 (iii) $S=0,\ P=\sqrt5$

$$f(x)=x^2-0x+\sqrt5$$
$$\boxed{x^2+\sqrt5}$$

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🔷 (iv) $S=1,\ P=1$

$$f(x)=x^2-x+1$$
$$\boxed{x^2-x+1}$$

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🔷 (v) $S=-\frac14,\ P=\frac14$

$$f(x)=x^2+\frac14x+\frac14$$

Multiplying by 4:

$$\boxed{4x^2+x+1}$$

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🔷 (vi) $S=4,\ P=1$

$$f(x)=x^2-4x+1$$
$$\boxed{x^2-4x+1}$$

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✅ Final Answers

Part	Quadratic Polynomial
(i)	$4x^2-x-4$
(ii)	$3x^2-3\sqrt2x+1$
(iii)	$x^2+\sqrt5$
(iv)	$x^2-x+1$
(v)	$4x^2+x+1$
(vi)	$x^2-4x+1$

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⭐ Important Formula

If zeroes are:

$$\alpha,\beta$$

then:

$$\alpha+\beta=S$$
$$\alpha\beta=P$$

Required quadratic polynomial:

$$\boxed{x^2-Sx+P}$$

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📚 Related Topics

• Number of Zeroes of a Polynomial from Graphs
• Prove that Given Numbers are Irrational
• Zeroes and Coefficients of Quadratic Polynomial