The Remainder Theorem and the Factor Theorem are fundamental concepts in algebra for analyzing polynomials. The Remainder Theorem states that when a polynomial \(f(x)\) is divided by \(x - c\), the remainder is equal to \(f(c)\). Building upon this, the Factor Theorem establishes a crucial link: \(x - c\) is a factor of the polynomial \(f(x)\) if and only if \(f(c) = 0\). These theorems allow for efficient factorization of higher-degree polynomials and identification of their roots without performing complex long division.
رقم الاختبار828
الصفالصف العاشر المتقدم
المادةرياضيات
الفصلالفصل الثالث
السنة الدراسية2025/2026
عدد الأسئلة10
إجمالي النقاط10
تاريخ الإضافة2026-04-21
الزيارات40
المعلم أو الناشرAmal Salman
اختر إجابة واحدة لكل سؤال. عند الاختيار ستظهر النتيجة فورًا: الأخضر صحيح، والأحمر خطأ، وسيظهر تفسير الإجابة مباشرة إن كان متوفرًا. وبعد آخر سؤال ستظهر الدرجة النهائية تلقائيًا.
Question 1
Points: 1
Which binomial is a factor of \(f(x) = x^3 - 6x^2 + 3x + 10\)?
Explanation
By the Factor Theorem, \(x + 1\) is a factor if \(f(-1) = 0\). Calculating \(f(-1) = (-1)^3 - 6(-1)^2 + 3(-1) + 10 = -1 - 6 - 3 + 10 = 0\). Therefore, \(x + 1\) is a factor.
Question 2
Points: 1
What is the remainder when \(a^3 - 4\) is divided by \(a + 2\)?
Explanation
According to the Remainder Theorem, the remainder when a polynomial \(P(a)\) is divided by \(a + 2\) is \(P(-2)\). Substituting \(a = -2\) into \(a^3 - 4\) gives \((-2)^3 - 4 = -8 - 4 = -12\).
Question 3
Points: 1
Which binomial is a factor of \(f(x) = x^3 + x^2 - 24x + 36\)?
Explanation
Using the Factor Theorem, testing \(x = -6\): \(f(-6) = (-6)^3 + (-6)^2 - 24(-6) + 36 = -216 + 36 + 144 + 36 = 0\). Since \(f(-6) = 0\), \(x + 6\) is a factor.
Question 4
Points: 1
What is the remainder \(R\) when the polynomial \(p(x)\) is divided by \(x - 5\)? \(p(x) = x^3 - 5x^2 + 2x - 10\)
Explanation
By the Remainder Theorem, the remainder is \(p(5)\). \(p(5) = 5^3 - 5(5)^2 + 2(5) - 10 = 125 - 125 + 10 - 10 = 0\).
Question 5
Points: 1
We know \(f(-1) = 0\) for the polynomial \(f(x) = x^3 - 6x^2 + 5x + 12\). What do we know is a factor of \(f(x)\)?
Explanation
According to the Factor Theorem, if \(f(c) = 0\), then \((x - c)\) is a factor. Given \(f(-1) = 0\), the factor is \(x - (-1) = x + 1\).
Question 6
Points: 1
We know \(f(4) = 0\) for \(f(x) = x^3 - 6x^2 + 5x + 12\). Factor \(f(x)\) completely using this information.
Explanation
Given \(f(4) = 0\), \(x - 4\) is a factor. Dividing the polynomial by \(x - 4\) gives \(x^2 - 2x - 3\), which factors into \((x + 1)(x - 3)\). Thus the complete factorization is \((x - 4)(x + 1)(x - 3)\).
Question 7
Points: 1
If \(f(x) = 3x^2 - 9x - 20\), find the value of \(f(5)\).
Find all the zeros, given that \(f(-3) = 0\). \(f(x) = 2x^3 + 5x^2 - 6x - 9\)
Explanation
Since \(-3\) is a zero, divide \(2x^3 + 5x^2 - 6x - 9\) by \(x + 3\) to get \(2x^2 - x - 3\). Factoring \(2x^2 - x - 3\) gives \((2x - 3)(x + 1)\), which yields zeros at \(3/2\) and \(-1\). The full set of zeros is \(-3, -1, 3/2\).
Question 9
Points: 1
Find all the factors of \(x^3 - 3x^2 - 4x + 12\) given that \(-2\) is a zero.
Explanation
Since \(-2\) is a zero, \(x + 2\) is a factor. Dividing \(x^3 - 3x^2 - 4x + 12\) by \(x + 2\) results in \(x^2 - 5x + 6\), which factors further into \((x - 2)(x - 3)\). The complete set of factors is \((x + 2)(x - 2)(x - 3)\).
Question 10
Points: 1
Find all the real zeros of the function \(f(x) = 2x^3 - 19x^2 + 38x + 24\) given that \(x - 4\) is a factor.
Explanation
Given \(x - 4\) is a factor, \(4\) is a zero. Using synthetic or long division to divide the polynomial by \(x - 4\) yields \(2x^2 - 11x - 6\). Factoring this quadratic gives \((2x + 1)(x - 6)\), providing zeros at \(-1/2\) and \(6\). The real zeros are \(4, -1/2, 6\).
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