اختبار إلكتروني: Solving Polynomial Equation, The Remainder and Factor
Polynomial equations are fundamental in algebra, involving terms with variables raised to non-negative integer powers. Understanding how to find remainders and determine factors is crucial for solving higher-degree equations. The Remainder Theorem states that the remainder of the division of a polynomial \(f(x)\) by a linear factor \((x - c)\) is simply \(f(c)\). Building on this, the Factor Theorem provides that if \(f(c) = 0\), then \((x - c)\) is a factor of the polynomial. Techniques such as synthetic division and factoring special patterns, like the difference of cubes or grouping, are essential tools for students to master when working with complex algebraic expressions.
رقم الاختبار829
الصفالصف العاشر المتقدم
المادةرياضيات
الفصلالفصل الثالث
السنة الدراسية2025/2026
عدد الأسئلة13
إجمالي النقاط13
تاريخ الإضافة2026-04-21
الزيارات44
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
What is the remainder when \(a^3 - 4\) is divided by \(a + 2\)?
Explanation
According to the Remainder Theorem, the remainder of dividing \(f(a) = a^3 - 4\) by \(a + 2\) is \(f(-2)\). Calculating: \((-2)^3 - 4 = -8 - 4 = -12\).
Question 2
Points: 1
Which binomial is a factor of \(f(x) = x^3 + x^2 - 24x + 36\)?
Explanation
Using the Factor Theorem, we check which value makes \(f(x) = 0\). For \(x + 6\), we check \(f(-6)\): \((-6)^3 + (-6)^2 - 24(-6) + 36 = -216 + 36 + 144 + 36 = 0\). Since \(f(-6) = 0\), \(x + 6\) is a factor.
Question 3
Points: 1
if \(f(x) = 3x^2 - 9x - 20\), find the value of \(f(5)\) using synthetic division.
Explanation
Using the Remainder Theorem, \(f(5)\) is the remainder of \(3x^2 - 9x - 20\) divided by \(x - 5\). Calculating directly: \(3(5)^2 - 9(5) - 20 = 3(25) - 45 - 20 = 75 - 45 - 20 = 10\).
Question 4
Points: 1
What are the three factors for \((x^3 + 7x^2 + 7x - 15) \div (x - 1)\)?
Explanation
First, divide \(x^3 + 7x^2 + 7x - 15\) by \(x - 1\) using synthetic division to get \(x^2 + 8x + 15\). Then factor the quadratic: \(x^2 + 8x + 15 = (x + 5)(x + 3)\). Thus, the three factors are \((x - 1)(x + 5)(x + 3)\).
Question 5
Points: 1
Factor this difference of cubes: \(x^3 - 343\)
Explanation
The formula for the difference of cubes is \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\). Here, \(a = x\) and \(b = 7\) (since \(7^3 = 343\)). Applying the formula gives \((x - 7)(x^2 + 7x + 49)\).
Question 6
Points: 1
If the polynomial \(x^2 - 5x + 9\) is divided by \((x - 3)\), then the remainder is
Explanation
By the Remainder Theorem, the remainder of dividing \(f(x) = x^2 - 5x + 9\) by \(x - 3\) is \(f(3)\). Calculating: \(3^2 - 5(3) + 9 = 9 - 15 + 9 = 3\).
Question 7
Points: 1
If \(f(x) = 5x^3 - 3x^2 + 1\), then the value of \(f\left(\frac{2}{5}\right)\) is
Divide using synthetic division: \((n^2 + 10n + 18)\) by \((n + 5)\)
Explanation
Using synthetic division with root \(-5\) and coefficients \([1, 10, 18]\): The first coefficient \(1\) drops down. \(-5 \times 1 = -5\). \(10 + (-5) = 5\). \(-5 \times 5 = -25\). \(18 + (-25) = -7\). The quotient is \(n + 5\) and the remainder is \(-7\).
Question 10
Points: 1
Factor: \(2x^3 + 54\)
Explanation
First, factor out the GCF, which is \(2\): \(2(x^3 + 27)\). Then, use the sum of cubes formula \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\) for \(x^3 + 3^3\), which gives \(2(x + 3)(x^2 - 3x + 9)\).
Question 11
Points: 1
Factor and solve: \(3n^3 - 4n^2 + 9n = 12\)
Explanation
Rearrange to \(3n^3 - 4n^2 + 9n - 12 = 0\). Factor by grouping: \(n^2(3n - 4) + 3(3n - 4) = 0\), which leads to \((n^2 + 3)(3n - 4) = 0\). Solving gives \(n = \frac{4}{3}\) and \(n^2 = -3 \Rightarrow n = \pm i\sqrt{3}\).
Question 12
Points: 1
Solve the inequality: \(x + 5 \leq 13\)
Explanation
Subtract \(5\) from both sides of the inequality: \(x \leq 13 - 5\), which results in \(x \leq 8\).
Question 13
Points: 1
Is \((x - 4)\) a factor of \((x^3 + x^2 - 16x - 16)\)?
Explanation
Using the Factor Theorem, check \(f(4)\): \((4)^3 + (4)^2 - 16(4) - 16 = 64 + 16 - 64 - 16 = 0\). Since the remainder is \(0\), \(x - 4\) is a factor.
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