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2026-01-01
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2026-02-21
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<p>Let α, β, and γ be the cubic polynomial \( ax^3 + bx^2 + cx + d = 0 \), where a ≠ 0 and a, b, and c are the coefficients of \(x^3\), \(x^2\), and x, and d is the constant term. Next, a cubic polynomial’s<a>product</a>of zeros is provided as follows:</p>
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<p>Let α, β, and γ be the cubic polynomial \( ax^3 + bx^2 + cx + d = 0 \), where a ≠ 0 and a, b, and c are the coefficients of \(x^3\), \(x^2\), and x, and d is the constant term. Next, a cubic polynomial’s<a>product</a>of zeros is provided as follows:</p>
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<p>Product of zeros of a cubic polynomial = \( -\dfrac{\text{constant term}}{\text{coefficient of } x^3} \).</p>
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<p>Product of zeros of a cubic polynomial = \( -\dfrac{\text{constant term}}{\text{coefficient of } x^3} \).</p>
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<p>⇒ \( \alpha \beta \gamma = -\frac{d}{a} \).</p>
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<p>⇒ \( \alpha \beta \gamma = -\frac{d}{a} \).</p>
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<p>Example: Determine the cubic polynomial \(\ kx^3 - 5x^2 - 12x + k = 0 \ \) . If the polynomial is \(\ kx^3 - 5x^2 - 12x + k = 0 \), then we have a = k, and d = k.</p>
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<p>Example: Determine the cubic polynomial \(\ kx^3 - 5x^2 - 12x + k = 0 \ \) . If the polynomial is \(\ kx^3 - 5x^2 - 12x + k = 0 \), then we have a = k, and d = k.</p>
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<p>Product of zeros = -(constant term)/(<a>coefficient</a>of \(x^3\))</p>
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<p>Product of zeros = -(constant term)/(<a>coefficient</a>of \(x^3\))</p>
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<p>⇒\(\ \alpha \beta \gamma = -\frac{k}{k} = -1 \ \)</p>
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<p>⇒\(\ \alpha \beta \gamma = -\frac{k}{k} = -1 \ \)</p>
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<p>Therefore, the product of the zeros of a cubic polynomial is -1.</p>
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<p>Therefore, the product of the zeros of a cubic polynomial is -1.</p>
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<p><strong>Sum of Zeros of a Cubic Polynomial</strong>Let α, β, and γ be the zeros of the cubic polynomial \(\ ax^3 + bx^2 + cx + d = 0 \ \) where a ≠ 0, where a, b, and c are the coefficients of \(x^3\),\(x^2\), and x and d is the constant term. Next, a cubic polynomial’s sum of zeros is expressed as follows:</p>
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<p><strong>Sum of Zeros of a Cubic Polynomial</strong>Let α, β, and γ be the zeros of the cubic polynomial \(\ ax^3 + bx^2 + cx + d = 0 \ \) where a ≠ 0, where a, b, and c are the coefficients of \(x^3\),\(x^2\), and x and d is the constant term. Next, a cubic polynomial’s sum of zeros is expressed as follows:</p>
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<p>\(\ \text{The sum of zeros of a cubic polynomial} = -\frac{\text{coefficient of } x^2}{\text{coefficient of } x^3} \ \)</p>
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<p>\(\ \text{The sum of zeros of a cubic polynomial} = -\frac{\text{coefficient of } x^2}{\text{coefficient of } x^3} \ \)</p>
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<p>⇒\(\ \alpha + \beta + \gamma = -\frac{b}{a} \ \).</p>
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<p>⇒\(\ \alpha + \beta + \gamma = -\frac{b}{a} \ \).</p>
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<p>Example: Determine the cubic polynomial \(\ 5x^3 - 15x^2 - 12x + 27 = 0 \ \).</p>
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<p>Example: Determine the cubic polynomial \(\ 5x^3 - 15x^2 - 12x + 27 = 0 \ \).</p>
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<p>\(\ \text{Sum of zeros of a cubic polynomial} = -\frac{\text{coefficient of } x^2}{\text{coefficient of } x^3} \ \) ⇒\(\ \alpha + \beta + \gamma = -\frac{b}{a} \ \).</p>
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<p>\(\ \text{Sum of zeros of a cubic polynomial} = -\frac{\text{coefficient of } x^2}{\text{coefficient of } x^3} \ \) ⇒\(\ \alpha + \beta + \gamma = -\frac{b}{a} \ \).</p>
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<p>If the polynomial is \(\ 5x^3 - 15x^2 - 12x + 27 = 0 \ \), then we have a= 5 and b = -15.</p>
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<p>If the polynomial is \(\ 5x^3 - 15x^2 - 12x + 27 = 0 \ \), then we have a= 5 and b = -15.</p>
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<p>Sum of zero = -ba</p>
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<p>Sum of zero = -ba</p>
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<p>⇒ \(\ \alpha + \beta + \gamma = -\left(-\frac{15}{5}\right) = 3 \ \)</p>
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<p>⇒ \(\ \alpha + \beta + \gamma = -\left(-\frac{15}{5}\right) = 3 \ \)</p>
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<p>Therefore, the given cubic polynomial’s sum of zeros is 3.</p>
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<p>Therefore, the given cubic polynomial’s sum of zeros is 3.</p>