Polynomials in one variable

The governing concept is finding the zeroes of a quadratic polynomial by solving p(x)=0 with the quadratic formula. Set x²−8x+10=0, where a=1, b=−8, and c=10. Then x=[−b±√(b²−4ac)]/(2a)=[8±√(64−40)]/2=[8±√24]/2. Since √24=√(4×6)=2√6, the roots become [8±2√6]/2=4±√6. Thus the zeroes are 4+√6 and 4−√6, so option A is correct. Option B uses an incorrect discriminant and fails to divide correctly. Option C has the wrong half of the constant term, while option D replaces √6 with √10 without justification. Substitution into the polynomial also verifies both roots.

For a quadratic polynomial ax² + bx + c with zeroes α and β, the relation α + β = −b/a holds, even when a coefficient is irrational. In p(x) = 2x² − 4√2x + 4, we have a = 2 and b = −4√2. Therefore α + β = −(−4√2)/2 = 4√2/2 = 2√2. The constant term c = 4 determines the product αβ = c/a, not the sum. Hence option A is correct. Option B results from forgetting to divide by the leading coefficient, option C comes from an incorrect division, and option D confuses the constant term with the sum. The standard coefficient-zero relation remains valid without any change.

For a monic quadratic with zeroes α and β, the standard relation is x² − (α + β)x + αβ. “Monic” means that the coefficient of x² is 1. The question gives α + β = 2√5 and αβ = 4. Substituting these values produces x² − 2√5x + 4, so option A is correct. Option B has the opposite sign in the middle term and would give a zero-sum of −2√5. Option C places the given sum and product in the wrong positions. Option D has an incorrect sign and also uses the wrong constant term. Irrational coefficients are permitted in a polynomial, so the presence of √5 does not invalidate option A.

The required quantity is the sum of the two given zeroes, so the radicals must first be simplified and then combined. Simplify √8 as √(4 × 2) = √4 × √2 = 2√2. Hence the sum is √2 + √8 = √2 + 2√2 = 3√2, making option A correct. Option C would result from incorrectly treating √8 as 3√2. Option B combines the radicands as if addition of radicals were multiplication, which is not a valid rule. Option D is also not the sum of the given zeroes. The key principle is that like surds can be added only after they have been expressed with the same square-free radicand; their coefficients are then added.

Use the perfect-square identity (x − a)² = x² − 2ax + a². Taking a = √2 gives (x − √2)² = x² − 2√2x + (√2)² = x² − 2√2x + 2, which is exactly the given polynomial. Therefore option A is correct. Option B expands to x² + 2√2x + 2, so its middle-term sign is wrong. Option C expands to x² − 4x + 4 and has different coefficients. Option D uses the difference-of-squares identity and gives x² − 2, which lacks the required middle term. The factorization also shows that √2 is a repeated zero, because the same linear factor occurs twice.

The governing concept is the difference-of-squares identity u²−v²=(u−v)(u+v). Take u=x^4 and v=9. Then x^8−81=(x^4)^2−9^2=(x^4−9)(x^4+9). Substituting this factorisation into the fraction gives [(x^4−9)(x^4+9)]/(x^4−9). The condition x^4≠9 ensures that the denominator is non-zero, so cancelling the common factor is valid. The simplified expression is x^4+9, making option B correct. Option A is the factor that cancels rather than the remaining result. Option C changes the power of x incorrectly, and option D uses a plus sign where the original numerator has a difference and does not follow from the identity.

In a polynomial in one variable, powers of the variable are non-negative integers. In exams, reject negative powers and roots of the variable.

The degree of a polynomial is the highest exponent of the variable with a non-zero coefficient. Here, the highest exponent is 5 and its coefficient is 4, so the degree is 5. The term 9 is a constant term and does not determine the degree. In an exam, identify the greatest exponent of the variable.

The term containing \(x^2\) is \(2x^2\), so its coefficient is 2. The number 7 is the coefficient of \(x^3\), while -1 is the coefficient of \(x\); neither is the coefficient of \(x^2\). In an exam, match the required power with its term and then read its numerical factor.

A constant term is a term that does not contain the variable \(x\). In the given polynomial, \(5x^4\) and \(-8x\) contain \(x\), whereas 11 does not; therefore, 11 is the constant term. The 4 in option D is the exponent, not a term. Exam tip: the standalone number in a polynomial is usually its constant term.

The terms of the polynomial are \(2x^3\), \(-5x^2\), \(4x\), and \(-1\), so it has 4 terms. Terms are separated by addition or subtraction signs; do not count coefficients or powers separately. Exam tip: count each positive or negative term individually.

To find \(p(2)\), substitute 2 for \(x\): \(p(2)=3(2)^2-2(2)+5=12-4+5=13\). Therefore, the correct answer is 13. The distractor 11 may result from combining the terms incorrectly. Exam tip: evaluate the exponent first, then perform multiplication and addition or subtraction.

Substituting \(x=-1\), we get \(q(-1)=(-1)^3-4(-1)^2+(-1)+6=-1-4-1+6=0\). Here, \((-1)^2=1\) but \((-1)^3=-1\), so the signs must be handled carefully. In an exam, evaluate each term separately before adding them.

The polynomial has three distinct terms: \\(8x^2\\), \\(-3x\\), and \\(1\\). Therefore, it is a trinomial. A binomial has exactly two terms, while a zero polynomial has all its coefficients equal to zero. In an exam, count the terms to classify a polynomial.

A polynomial whose all coefficients are zero is called the zero polynomial; therefore, \(p(x)=0\) is the zero polynomial. It is not classified as a non-zero constant polynomial, and its degree is considered undefined. Exam tip: remember that the degree of the zero polynomial is undefined.

A non-zero constant polynomial contains no variable term, so it can be written as \(12x^0\); therefore, its degree is 0. The value 12 is the constant term, not the degree. Remember that the degree of the zero polynomial is generally undefined, but this polynomial is non-zero.

The term (0x^3) is ineffective, and the highest non-zero power is (4). Ignore terms with zero coefficient.

In \(4x^3-2x+7\), the powers of \(x\) are 3, 1 and 0, all non-negative integers; hence it is a polynomial in one variable. \(3/x=3x^{-1}\) has a negative exponent. Exam tip: check that every variable exponent is 0, 1, 2, ... .

In the term
kx
, the coefficient of
x
is
k
. Therefore, if the coefficient of
x
is 5, then
k=5
. The value 8 is the constant term, not the coefficient of
x
. Exam tip: identify the factor multiplying the specified variable to find its coefficient.

To find \(p(3)\), substitute 3 for \(x\): \(p(3)=3^2-9=9-9=0\). Hence, the correct answer is 0. The value 9 is only \(3^2\); the subtraction of 9 must also be performed. In exams, directly substitute the given value of the variable into the polynomial.

A zero of a polynomial is a value that makes the polynomial equal to zero. For option A, \(p(2)=2-2=0\), so \(x=2\) is its zero. In contrast, option B gives \(p(2)=2+2=4\), so it is not correct. Exam tip: Substitute the given value into each polynomial and check whether the result is zero.

Substitute \(x=2\) into the polynomial: \(p(2)=2^2+2(2)-8=4+4-8=0\). Therefore, the correct answer is 0. Since the polynomial’s value is zero at \(x=2\), 2 is also a zero of the polynomial. Exam tip: substitute the given value carefully before simplifying.

The degree of a polynomial is the highest exponent of the variable whose coefficient is non-zero. In \(ax+b\), the condition \(a\ne0\) ensures that the term \(ax\) is present, so the highest power of \(x\) is 1. Therefore, its degree is 1. Option 0 applies to a non-zero constant polynomial, not a linear polynomial. Exam tip: The general form of a linear polynomial is \(ax+b\) with \(a\ne0\).

The term with the highest power in the polynomial is \(9y^4\). Its coefficient is 9, so the leading coefficient is 9. The value -1 in option B is the coefficient of the \(y^2\) term, not the leading term. Exam tip: identify the highest-degree term first, then take its coefficient.

The term containing \\(x^3\\) is \\(4x^3\\), so its coefficient is 4. The terms \\(-2x^2\\), \\(7x\\), and \\(-1\\) belong to different powers or are the constant term. Exam tip: identify the number multiplying the required power of the variable.