With rational coefficients \(a+\sqrt{b}\) is accompanied by \(a-\sqrt{b}\). In exams identify conjugate zeroes quickly.
Step 2
Why this answer is correct
The correct answer is A. \(4-\sqrt{11}\). With rational coefficients \(a+\sqrt{b}\) is accompanied by \(a-\sqrt{b}\). In exams identify conjugate zeroes quickly.
Step 3
Exam Tip
परिमेय गुणांकों में \(a+\sqrt{b}\) के साथ \(a-\sqrt{b}\) भी शून्यक होता है। परीक्षा में संयुग्मी शून्यक तुरंत पहचानें।
The companion zero is \(2-\sqrt{3}\), so the factor is (x-\(2-\sqrt{3}\)). In exams remember the relation between a zero and factor as \(x-\alpha\).
Step 2
Why this answer is correct
The correct answer is A. (x-\(2-\sqrt{3}\)). The companion zero is \(2-\sqrt{3}\), so the factor is (x-\(2-\sqrt{3}\)). In exams remember the relation between a zero and factor as \(x-\alpha\).
Step 3
Exam Tip
साथी शून्यक \(2-\sqrt{3}\) होगा, इसलिए गुणनखंड (x-\(2-\sqrt{3}\)) है। परीक्षा में शून्यक और गुणनखंड का संबंध \(x-\alpha\) याद रखें।
The sum is (2) and the product is (1-6=-5), so the polynomial is \(x^2-2x-5\). In exams use \(a^2-b^2\) for the product.
Step 2
Why this answer is correct
The correct answer is A. \(x^2-2x-5\). The sum is (2) and the product is (1-6=-5), so the polynomial is \(x^2-2x-5\). In exams use \(a^2-b^2\) for the product.
Step 3
Exam Tip
योग (2) और गुणनफल (1-6=-5) है, इसलिए बहुपद \(x^2-2x-5\) है। परीक्षा में गुणनफल में \(a^2-b^2\) लगाएं।
The constant term is the product, and (\(4+\sqrt{11}\)\(4-\sqrt{11}\)=16-11=5). In conjugate products, the irrational middle part cancels.
Step 2
Why this answer is correct
The correct answer is A. (5). The constant term is the product, and (\(4+\sqrt{11}\)\(4-\sqrt{11}\)=16-11=5). In conjugate products, the irrational middle part cancels.
Step 3
Exam Tip
स्थिर पद गुणनफल है और (\(4+\sqrt{11}\)\(4-\sqrt{11}\)=16-11=5)। संयुग्मी गुणनफल में बीच का अपरिमेय भाग हट जाता है।
The other zero is \(6+2\sqrt{5}\). The sum is (12) and product is (36-20=16), so the polynomial is \(x^2-12x+16\).
Step 2
Why this answer is correct
The correct answer is A. \(x^2-12x+16\). The other zero is \(6+2\sqrt{5}\). The sum is (12) and product is (36-20=16), so the polynomial is \(x^2-12x+16\).
Step 3
Exam Tip
दूसरा शून्यक \(6+2\sqrt{5}\) होगा। योग (12) और गुणनफल (36-20=16), इसलिए बहुपद \(x^2-12x+16\) है।
A. योग (2a) और गुणनफल \(a^2-5\) हैं/Sum is (2a) and product is \(a^2-5\)
Step 1
Concept
(\(a+\sqrt{5}\)+\(a-\sqrt{5}\)=2a) and (\(a+\sqrt{5}\)\(a-\sqrt{5}\)=a-2-5). These match the polynomial.
Step 2
Why this answer is correct
The correct answer is A. योग (2a) और गुणनफल \(a^2-5\) हैं / Sum is (2a) and product is \(a^2-5\). (\(a+\sqrt{5}\)+\(a-\sqrt{5}\)=2a) and (\(a+\sqrt{5}\)\(a-\sqrt{5}\)=a-2-5). These match the polynomial.
Step 3
Exam Tip
(\(a+\sqrt{5}\)+\(a-\sqrt{5}\)=2a) और (\(a+\sqrt{5}\)\(a-\sqrt{5}\)=a-2-5)। यही बहुपद से मेल खाता है।
For a quadratic with rational coefficients, \(a-\sqrt{b}\) accompanies \(a+\sqrt{b}\). Remember this as the conjugate-zero rule.
Step 2
Why this answer is correct
The correct answer is A. \(3-\sqrt{5}\). For a quadratic with rational coefficients, \(a-\sqrt{b}\) accompanies \(a+\sqrt{b}\). Remember this as the conjugate-zero rule.
Step 3
Exam Tip
परिमेय गुणांकों वाले द्विघात में \(a+\sqrt{b}\) के साथ \(a-\sqrt{b}\) भी शून्यक होता है। परीक्षा में इसे संयुग्मी शून्यक नियम की तरह याद रखें।
