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2026-01-01
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2026-02-28
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<p>According to the rational root theorem, for the root of a<a>polynomial</a>to be a rational<a>number</a>, the<a>numerator and denominator</a>must be<a>factors</a>of the<a>constant</a><a>term</a>and leading<a>coefficient</a>, respectively. The leading coefficient is the coefficient of the term that has the highest power of the variable.</p>
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<p>According to the rational root theorem, for the root of a<a>polynomial</a>to be a rational<a>number</a>, the<a>numerator and denominator</a>must be<a>factors</a>of the<a>constant</a><a>term</a>and leading<a>coefficient</a>, respectively. The leading coefficient is the coefficient of the term that has the highest power of the variable.</p>
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<p>For a<a>polynomial</a>:</p>
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<p>For a<a>polynomial</a>:</p>
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<p>\(P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0 \),</p>
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<p>\(P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0 \),</p>
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<p>\(a_n\) denotes the leading coefficient and a0 denotes the constant term. </p>
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<p>\(a_n\) denotes the leading coefficient and a0 denotes the constant term. </p>
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<p><strong>Rational Root Theorem Statement</strong></p>
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<p><strong>Rational Root Theorem Statement</strong></p>
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<p>The rational root theorem states: </p>
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<p>The rational root theorem states: </p>
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<p>If a polynomial equation \(p(x) = anxn + an-1xn-1 + . . . + a1x +a0\), has a rational root \(\frac{p}{q} \) in its lowest terms, then: </p>
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<p>If a polynomial equation \(p(x) = anxn + an-1xn-1 + . . . + a1x +a0\), has a rational root \(\frac{p}{q} \) in its lowest terms, then: </p>
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<ul><li>p must be a factor (divisor) of the constant term a₀, and </li>
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<ul><li>p must be a factor (divisor) of the constant term a₀, and </li>
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<li>q must be a factor (divisor) of the leading coefficient, aₙ. </li>
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<li>q must be a factor (divisor) of the leading coefficient, aₙ. </li>
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</ul><p>Any possible rational solution to the polynomial must be formed by dividing a factor of the constant term by a factor of the leading coefficient.</p>
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</ul><p>Any possible rational solution to the polynomial must be formed by dividing a factor of the constant term by a factor of the leading coefficient.</p>
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<p>Therefore, possible rational roots of \(\frac{p}{q} \) are of the form \(\frac{\text{factors of } a_0}{\text{factors of } a_n} \).</p>
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<p>Therefore, possible rational roots of \(\frac{p}{q} \) are of the form \(\frac{\text{factors of } a_0}{\text{factors of } a_n} \).</p>
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<p><strong>Rational Root Theorem Definition</strong></p>
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<p><strong>Rational Root Theorem Definition</strong></p>
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<p>The rational root theorem is a mathematical rule for finding all possible rational zeroes (or roots) of a polynomial equation with integer coefficients. It states that if a polynomial has a rational root written in the form \(\frac{p}{q}\) in its lowest terms, then the numerator p must be a factor of the constant term, and the denominator q must be a factor of the leading coefficient.</p>
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<p>The rational root theorem is a mathematical rule for finding all possible rational zeroes (or roots) of a polynomial equation with integer coefficients. It states that if a polynomial has a rational root written in the form \(\frac{p}{q}\) in its lowest terms, then the numerator p must be a factor of the constant term, and the denominator q must be a factor of the leading coefficient.</p>
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<p><strong>Rational Root Theorem Examples</strong> </p>
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<p><strong>Rational Root Theorem Examples</strong> </p>
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<p><strong>Example 1:</strong>Find the possible rational roots of \(p(x) = x^3-6x^2 + 11x - 6\). Here, the constant term a0 = -6. Leading coefficient an = 1. Factors of -6 are ±1, ±2, ±3, ±6. Factors of 1 are ±1. The possible rational roots are ±1, ±2, ±3, and ±6. </p>
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<p><strong>Example 1:</strong>Find the possible rational roots of \(p(x) = x^3-6x^2 + 11x - 6\). Here, the constant term a0 = -6. Leading coefficient an = 1. Factors of -6 are ±1, ±2, ±3, ±6. Factors of 1 are ±1. The possible rational roots are ±1, ±2, ±3, and ±6. </p>
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<p><strong>Example 2:</strong>Find the possible rational roots of \(p(x) = 2x^3 + x^2 - 7x - 6\). Here, constant term a0 = -6. Leading coefficient an = 2. Factors of -6 are ±1, 2, ±3, and ±6. Factors of 2 are ±1 and ±2. Therefore, the possible rational roots are ±1, ±2, ±3, ±6, ±1/2, ±3/2</p>
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<p><strong>Example 2:</strong>Find the possible rational roots of \(p(x) = 2x^3 + x^2 - 7x - 6\). Here, constant term a0 = -6. Leading coefficient an = 2. Factors of -6 are ±1, 2, ±3, and ±6. Factors of 2 are ±1 and ±2. Therefore, the possible rational roots are ±1, ±2, ±3, ±6, ±1/2, ±3/2</p>
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